L(s) = 1 | + 6·5-s + 3·7-s + 3·11-s + 6·13-s + 12·17-s + 6·19-s − 15·23-s + 18·25-s − 27-s − 12·29-s − 6·31-s + 18·35-s − 12·37-s − 18·41-s − 18·43-s − 3·47-s + 15·49-s − 24·53-s + 18·55-s + 18·59-s − 9·61-s + 36·65-s − 6·67-s − 18·71-s + 21·73-s + 9·77-s + 6·79-s + ⋯ |
L(s) = 1 | + 2.68·5-s + 1.13·7-s + 0.904·11-s + 1.66·13-s + 2.91·17-s + 1.37·19-s − 3.12·23-s + 18/5·25-s − 0.192·27-s − 2.22·29-s − 1.07·31-s + 3.04·35-s − 1.97·37-s − 2.81·41-s − 2.74·43-s − 0.437·47-s + 15/7·49-s − 3.29·53-s + 2.42·55-s + 2.34·59-s − 1.15·61-s + 4.46·65-s − 0.733·67-s − 2.13·71-s + 2.45·73-s + 1.02·77-s + 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.019554633\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.019554633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 - 6 T - 12 T^{2} + 169 T^{3} - 12 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
good | 5 | \( 1 - 6 T + 18 T^{2} - 9 p T^{3} + 81 T^{4} - 87 T^{5} + 109 T^{6} - 87 p T^{7} + 81 p^{2} T^{8} - 9 p^{4} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 3 T - 6 T^{2} + 5 T^{3} + 45 T^{4} + 108 T^{5} - 705 T^{6} + 108 p T^{7} + 45 p^{2} T^{8} + 5 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 3 T - 6 T^{2} + 81 T^{3} - 129 T^{4} - 318 T^{5} + 3067 T^{6} - 318 p T^{7} - 129 p^{2} T^{8} + 81 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 6 T + 42 T^{2} - 154 T^{3} + 864 T^{4} - 3132 T^{5} + 14823 T^{6} - 3132 p T^{7} + 864 p^{2} T^{8} - 154 p^{3} T^{9} + 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 12 T + 54 T^{2} - 81 T^{3} + 9 p T^{4} - 3999 T^{5} + 27073 T^{6} - 3999 p T^{7} + 9 p^{3} T^{8} - 81 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 15 T + 144 T^{2} + 1116 T^{3} + 7191 T^{4} + 40695 T^{5} + 205741 T^{6} + 40695 p T^{7} + 7191 p^{2} T^{8} + 1116 p^{3} T^{9} + 144 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 12 T + 72 T^{2} + 234 T^{3} + 1404 T^{4} + 15492 T^{5} + 119215 T^{6} + 15492 p T^{7} + 1404 p^{2} T^{8} + 234 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 6 T - 42 T^{2} - 238 T^{3} + 1548 T^{4} + 4284 T^{5} - 39009 T^{6} + 4284 p T^{7} + 1548 p^{2} T^{8} - 238 p^{3} T^{9} - 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 6 T + 102 T^{2} + 427 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 18 T + 189 T^{2} + 1683 T^{3} + 12609 T^{4} + 90819 T^{5} + 629218 T^{6} + 90819 p T^{7} + 12609 p^{2} T^{8} + 1683 p^{3} T^{9} + 189 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 18 T + 270 T^{2} + 3044 T^{3} + 28584 T^{4} + 235404 T^{5} + 1630749 T^{6} + 235404 p T^{7} + 28584 p^{2} T^{8} + 3044 p^{3} T^{9} + 270 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 3 T - 9 T^{2} + 513 T^{3} - 324 T^{4} - 5370 T^{5} + 215173 T^{6} - 5370 p T^{7} - 324 p^{2} T^{8} + 513 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 24 T + 243 T^{2} + 1035 T^{3} - 3843 T^{4} - 111855 T^{5} - 1075634 T^{6} - 111855 p T^{7} - 3843 p^{2} T^{8} + 1035 p^{3} T^{9} + 243 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 18 T + 144 T^{2} - 603 T^{3} - 4707 T^{4} + 98199 T^{5} - 856835 T^{6} + 98199 p T^{7} - 4707 p^{2} T^{8} - 603 p^{3} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 9 T + 36 T^{2} - 628 T^{3} - 2682 T^{4} - 1089 T^{5} + 320655 T^{6} - 1089 p T^{7} - 2682 p^{2} T^{8} - 628 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 6 T + 12 T^{2} + 338 T^{3} - 1512 T^{4} - 49140 T^{5} - 303915 T^{6} - 49140 p T^{7} - 1512 p^{2} T^{8} + 338 p^{3} T^{9} + 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 18 T + 270 T^{2} + 2736 T^{3} + 25812 T^{4} + 198576 T^{5} + 1613305 T^{6} + 198576 p T^{7} + 25812 p^{2} T^{8} + 2736 p^{3} T^{9} + 270 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 21 T + 156 T^{2} - 4 T^{3} - 6993 T^{4} + 46629 T^{5} - 275871 T^{6} + 46629 p T^{7} - 6993 p^{2} T^{8} - 4 p^{3} T^{9} + 156 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 6 T + 78 T^{2} + 800 T^{3} + 3348 T^{4} - 23760 T^{5} + 1487457 T^{6} - 23760 p T^{7} + 3348 p^{2} T^{8} + 800 p^{3} T^{9} + 78 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 15 T + 12 T^{2} + 1791 T^{3} - 9705 T^{4} - 103974 T^{5} + 2042971 T^{6} - 103974 p T^{7} - 9705 p^{2} T^{8} + 1791 p^{3} T^{9} + 12 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 15 T + 108 T^{2} - 396 T^{3} - 6489 T^{4} + 85359 T^{5} - 603863 T^{6} + 85359 p T^{7} - 6489 p^{2} T^{8} - 396 p^{3} T^{9} + 108 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 9 T + 90 T^{2} - 826 T^{3} - 243 T^{4} + 27189 T^{5} - 54627 T^{6} + 27189 p T^{7} - 243 p^{2} T^{8} - 826 p^{3} T^{9} + 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.63040469618717886562707883473, −6.43130137451659089291134771396, −6.13380945525004501970131775081, −6.08240680409643896140716212314, −5.93770865525428713565115711654, −5.77389603712965229871163452022, −5.77139385449611520013451426479, −5.44386990900650385503553746135, −5.22282740074355034629973936268, −5.02841446485067326199502239102, −4.85420373155565804673294699728, −4.67459349767027361157900609009, −4.55075515897587752660569666905, −3.72274856455871094989509804414, −3.65938114403136231413437650911, −3.54099697526627965675639475883, −3.52780249537256154586115387943, −3.28360601466425857467801560194, −3.06394800953029626759354204266, −2.04929780112509732001629821488, −2.01093043126492304231863693633, −1.99206244708200306053467759949, −1.60618311660037267623317490885, −1.59752548520600168387842425599, −0.983818989644761595338365006099,
0.983818989644761595338365006099, 1.59752548520600168387842425599, 1.60618311660037267623317490885, 1.99206244708200306053467759949, 2.01093043126492304231863693633, 2.04929780112509732001629821488, 3.06394800953029626759354204266, 3.28360601466425857467801560194, 3.52780249537256154586115387943, 3.54099697526627965675639475883, 3.65938114403136231413437650911, 3.72274856455871094989509804414, 4.55075515897587752660569666905, 4.67459349767027361157900609009, 4.85420373155565804673294699728, 5.02841446485067326199502239102, 5.22282740074355034629973936268, 5.44386990900650385503553746135, 5.77139385449611520013451426479, 5.77389603712965229871163452022, 5.93770865525428713565115711654, 6.08240680409643896140716212314, 6.13380945525004501970131775081, 6.43130137451659089291134771396, 6.63040469618717886562707883473
Plot not available for L-functions of degree greater than 10.