| L(s) = 1 | + 4·3-s − 12·5-s − 36·7-s − 30·9-s − 20·11-s − 84·13-s − 48·15-s + 34·17-s + 32·19-s − 144·21-s + 44·23-s − 94·25-s − 196·27-s − 396·29-s + 116·31-s − 80·33-s + 432·35-s − 140·37-s − 336·39-s − 60·41-s + 640·43-s + 360·45-s − 496·47-s + 394·49-s + 136·51-s + 236·53-s + 240·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 1.07·5-s − 1.94·7-s − 1.11·9-s − 0.548·11-s − 1.79·13-s − 0.826·15-s + 0.485·17-s + 0.386·19-s − 1.49·21-s + 0.398·23-s − 0.751·25-s − 1.39·27-s − 2.53·29-s + 0.672·31-s − 0.422·33-s + 2.08·35-s − 0.622·37-s − 1.37·39-s − 0.228·41-s + 2.26·43-s + 1.19·45-s − 1.53·47-s + 1.14·49-s + 0.373·51-s + 0.611·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 12 T + 238 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 36 T + 902 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 20 T + 734 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 84 T + 4430 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32 T - 3354 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 17318 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 396 T + 79870 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 116 T + 46518 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 140 T + 9006 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 60 T + 138550 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 640 T + 257526 T^{2} - 640 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 496 T + 225950 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 236 T + 302270 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 576 T + 404950 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 348 T + 228446 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1528 T + 1182150 T^{2} - 1528 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 876 T + 634054 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 380 T + 790902 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 172 T + 634326 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1024 T + 792806 T^{2} + 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 844 T + 265334 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 36 T + 1788038 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.46764324905871366406643248770, −12.11865127566980063387473303895, −11.40601723683768739651577938518, −11.23176205927366584067981330293, −10.06677886992225827536822004987, −9.953914877173093166775737652918, −9.231328933252150300014406119221, −9.046126559001149467984437485107, −7.931988669057157999926295339642, −7.87932884677054957333475707791, −7.19694506873248179453620676188, −6.63430043099024741022471979425, −5.59869375437530191348709478931, −5.41401985460512562094356725340, −4.11097259861506740270834759732, −3.52491421178731039093130966559, −2.93688891960522502344082134268, −2.38828464470879361637880457965, 0, 0,
2.38828464470879361637880457965, 2.93688891960522502344082134268, 3.52491421178731039093130966559, 4.11097259861506740270834759732, 5.41401985460512562094356725340, 5.59869375437530191348709478931, 6.63430043099024741022471979425, 7.19694506873248179453620676188, 7.87932884677054957333475707791, 7.931988669057157999926295339642, 9.046126559001149467984437485107, 9.231328933252150300014406119221, 9.953914877173093166775737652918, 10.06677886992225827536822004987, 11.23176205927366584067981330293, 11.40601723683768739651577938518, 12.11865127566980063387473303895, 12.46764324905871366406643248770
