| L(s) = 1 | − 4·3-s − 8·5-s − 22·7-s − 3·9-s + 28·11-s + 30·13-s + 32·15-s − 51·17-s − 80·19-s + 88·21-s − 142·23-s − 267·25-s + 76·27-s − 456·29-s − 230·31-s − 112·33-s + 176·35-s + 356·37-s − 120·39-s − 294·41-s − 556·43-s + 24·45-s − 640·47-s − 407·49-s + 204·51-s + 302·53-s − 224·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.715·5-s − 1.18·7-s − 1/9·9-s + 0.767·11-s + 0.640·13-s + 0.550·15-s − 0.727·17-s − 0.965·19-s + 0.914·21-s − 1.28·23-s − 2.13·25-s + 0.541·27-s − 2.91·29-s − 1.33·31-s − 0.590·33-s + 0.849·35-s + 1.58·37-s − 0.492·39-s − 1.11·41-s − 1.97·43-s + 0.0795·45-s − 1.98·47-s − 1.18·49-s + 0.560·51-s + 0.782·53-s − 0.549·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20123648 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20123648 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + 19 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 8 T + 331 T^{2} + 2032 T^{3} + 331 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 22 T + 891 T^{2} + 14300 T^{3} + 891 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 2627 T^{2} - 69844 T^{3} + 2627 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 30 T + 5119 T^{2} - 141212 T^{3} + 5119 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 80 T + 15945 T^{2} + 757312 T^{3} + 15945 p^{3} T^{4} + 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 142 T + 20731 T^{2} + 1854884 T^{3} + 20731 p^{3} T^{4} + 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 456 T + 127075 T^{2} + 23761392 T^{3} + 127075 p^{3} T^{4} + 456 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 230 T + 77787 T^{2} + 13785468 T^{3} + 77787 p^{3} T^{4} + 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 356 T + 133995 T^{2} - 29888184 T^{3} + 133995 p^{3} T^{4} - 356 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 294 T + 120199 T^{2} + 38886804 T^{3} + 120199 p^{3} T^{4} + 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 556 T + 289617 T^{2} + 81141512 T^{3} + 289617 p^{3} T^{4} + 556 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 640 T + 396797 T^{2} + 134564608 T^{3} + 396797 p^{3} T^{4} + 640 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 302 T + 293171 T^{2} - 71759636 T^{3} + 293171 p^{3} T^{4} - 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 636 T + 514369 T^{2} + 211823016 T^{3} + 514369 p^{3} T^{4} + 636 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 84 T + 556531 T^{2} + 31340024 T^{3} + 556531 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 1008 T + 967329 T^{2} + 607104160 T^{3} + 967329 p^{3} T^{4} + 1008 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 402 T + 483859 T^{2} - 12894428 T^{3} + 483859 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 838 T + 1394903 T^{2} - 671950004 T^{3} + 1394903 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 594 T + 357843 T^{2} + 156405492 T^{3} + 357843 p^{3} T^{4} - 594 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2396 T + 3204249 T^{2} - 2882084008 T^{3} + 3204249 p^{3} T^{4} - 2396 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 170 T + 1042603 T^{2} - 206881916 T^{3} + 1042603 p^{3} T^{4} + 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 270 T + 2151919 T^{2} + 286220420 T^{3} + 2151919 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.94076462691683665047061668054, −10.34115200803620225763799917905, −10.07597308605843844692440245967, −9.654172052372214001328624595919, −9.456899395693986915153238815679, −9.160567057324715243015193041935, −8.924195929553904985738525965411, −8.230423205343344515355480256609, −8.011328295153391208536212759736, −7.87933435992906322972185563031, −7.39541081230116923429254883473, −6.86376739299499030341353505875, −6.58971475348177821121418228799, −6.16582150560964598960285167668, −6.09583738763501111637602065863, −5.81710138455725822863304685397, −5.03757430943605731463273866715, −4.93546787514573055661494939720, −4.17991056957941028672213158537, −3.80529691290709977486016594149, −3.50439886027380408681771841824, −3.47660791316925268020536526913, −2.37745099399638889191827617779, −1.85328112104661425664183214095, −1.52653793537905168706477147795, 0, 0, 0,
1.52653793537905168706477147795, 1.85328112104661425664183214095, 2.37745099399638889191827617779, 3.47660791316925268020536526913, 3.50439886027380408681771841824, 3.80529691290709977486016594149, 4.17991056957941028672213158537, 4.93546787514573055661494939720, 5.03757430943605731463273866715, 5.81710138455725822863304685397, 6.09583738763501111637602065863, 6.16582150560964598960285167668, 6.58971475348177821121418228799, 6.86376739299499030341353505875, 7.39541081230116923429254883473, 7.87933435992906322972185563031, 8.011328295153391208536212759736, 8.230423205343344515355480256609, 8.924195929553904985738525965411, 9.160567057324715243015193041935, 9.456899395693986915153238815679, 9.654172052372214001328624595919, 10.07597308605843844692440245967, 10.34115200803620225763799917905, 10.94076462691683665047061668054
