| L(s) = 1 | + 2·3-s − 4·5-s − 44·7-s − 51·9-s − 40·11-s − 20·13-s − 8·15-s + 46·17-s − 38·19-s − 88·21-s + 220·23-s + 134·25-s − 158·27-s − 84·29-s + 84·31-s − 80·33-s + 176·35-s + 304·37-s − 40·39-s + 168·41-s + 372·43-s + 204·45-s − 584·47-s + 859·49-s + 92·51-s + 484·53-s + 160·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.357·5-s − 2.37·7-s − 1.88·9-s − 1.09·11-s − 0.426·13-s − 0.137·15-s + 0.656·17-s − 0.458·19-s − 0.914·21-s + 1.99·23-s + 1.07·25-s − 1.12·27-s − 0.537·29-s + 0.486·31-s − 0.422·33-s + 0.849·35-s + 1.35·37-s − 0.164·39-s + 0.639·41-s + 1.31·43-s + 0.675·45-s − 1.81·47-s + 2.50·49-s + 0.252·51-s + 1.25·53-s + 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2803532505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2803532505\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p^{3} T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 4 T - 118 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 44 T + 1077 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 40 T + 2690 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 20 T + 3657 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 46 T + 259 p T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 220 T + 1483 p T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 16969 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 84 T + 31214 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 304 T + 121062 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 168 T + 107698 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 372 T + 193238 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 584 T + 239342 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 484 T + 255041 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1074 T + 697639 T^{2} - 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 88 T + 437670 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1430 T + 1039839 T^{2} + 1430 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1848 T + 1569226 T^{2} + 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1294 T + 1190691 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 832 T + 593322 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 528 T + 459970 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1844 T + 1750754 T^{2} - 1844 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 364 T + 1606998 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27515208820760892419505755222, −10.16264303250350282689649338453, −9.459081276103157437696641763516, −9.153534820606796760732425292215, −8.641733692350637995433035944304, −8.597049310500871883789212079056, −7.65732181937602847318396821126, −7.44000982277327330151422679509, −6.94859454616858024469493404027, −6.33621007159613436769203307990, −5.80664208211596969154318344171, −5.73225581193242366083231595119, −4.93683904627857239386392756441, −4.36550849529769343197456238373, −3.47035502050648098825825461420, −3.09919009175409011763149025061, −2.75177402562111896270057726427, −2.54833296694205446796372393919, −0.956057485815152916947758909253, −0.18056801431116566529859879655,
0.18056801431116566529859879655, 0.956057485815152916947758909253, 2.54833296694205446796372393919, 2.75177402562111896270057726427, 3.09919009175409011763149025061, 3.47035502050648098825825461420, 4.36550849529769343197456238373, 4.93683904627857239386392756441, 5.73225581193242366083231595119, 5.80664208211596969154318344171, 6.33621007159613436769203307990, 6.94859454616858024469493404027, 7.44000982277327330151422679509, 7.65732181937602847318396821126, 8.597049310500871883789212079056, 8.641733692350637995433035944304, 9.153534820606796760732425292215, 9.459081276103157437696641763516, 10.16264303250350282689649338453, 10.27515208820760892419505755222
