| L(s) = 1 | − 16·2-s + 192·4-s + 114·5-s + 280·7-s − 2.04e3·8-s − 1.82e3·10-s + 5.18e3·11-s − 6.66e3·13-s − 4.48e3·14-s + 2.04e4·16-s + 3.65e4·17-s + 6.47e3·19-s + 2.18e4·20-s − 8.29e4·22-s − 1.29e4·23-s − 7.70e4·25-s + 1.06e5·26-s + 5.37e4·28-s + 1.60e4·29-s − 1.60e5·31-s − 1.96e5·32-s − 5.84e5·34-s + 3.19e4·35-s − 2.86e5·37-s − 1.03e5·38-s − 2.33e5·40-s − 5.38e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.407·5-s + 0.308·7-s − 1.41·8-s − 0.576·10-s + 1.17·11-s − 0.841·13-s − 0.436·14-s + 5/4·16-s + 1.80·17-s + 0.216·19-s + 0.611·20-s − 1.66·22-s − 0.222·23-s − 0.986·25-s + 1.18·26-s + 0.462·28-s + 0.122·29-s − 0.965·31-s − 1.06·32-s − 2.54·34-s + 0.125·35-s − 0.931·37-s − 0.306·38-s − 0.576·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.046875971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046875971\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 114 T + 18011 p T^{2} - 114 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 40 p T + 555582 T^{2} - 40 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5184 T + 27915142 T^{2} - 5184 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6662 T + 19857231 T^{2} + 6662 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 36510 T + 886979635 T^{2} - 36510 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6472 T + 1580438790 T^{2} - 6472 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12960 T + 6740530894 T^{2} + 12960 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 16074 T + 27847652863 T^{2} - 16074 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 160136 T + 60626118030 T^{2} + 160136 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 286958 T + 70225805703 T^{2} + 286958 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 538284 T + 238942946710 T^{2} + 538284 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1770896 T + 1319815433094 T^{2} + 1770896 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 279960 T + 675632927470 T^{2} - 279960 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2373852 T + 3297400780606 T^{2} - 2373852 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 690336 T + 36525190438 T^{2} - 690336 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3688886 T + 9632325380127 T^{2} + 3688886 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 983096 T + 6122082589350 T^{2} + 983096 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2776920 T + 17581276612798 T^{2} - 2776920 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9891094 T + 46451222820267 T^{2} - 9891094 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5078336 T + 43554395985246 T^{2} + 5078336 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8946312 T + 49317386349190 T^{2} - 8946312 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 682566 T + 64419777386971 T^{2} - 682566 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18464252 T + 245717615436102 T^{2} + 18464252 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.59942638354748654326905038679, −11.45570590780023874320557567955, −10.41021592605496545386155357545, −10.24049866181152910309926682308, −9.686682845696963045888210594374, −9.464499930887899250102294880192, −8.622916873325483920549799157987, −8.426766434051111891829236070031, −7.52502321592806223024002675380, −7.44008953909143964250731261287, −6.58617142680172883800545352074, −6.22930630098161077520130002615, −5.30663453183460788678526653635, −5.04371999856361799105281099036, −3.57807503794106469640820012141, −3.50052550588172430899741801395, −2.29404092134035447173543860672, −1.69357054407533087992533680894, −1.26078895091058140871697380014, −0.35663786146726449412626147086,
0.35663786146726449412626147086, 1.26078895091058140871697380014, 1.69357054407533087992533680894, 2.29404092134035447173543860672, 3.50052550588172430899741801395, 3.57807503794106469640820012141, 5.04371999856361799105281099036, 5.30663453183460788678526653635, 6.22930630098161077520130002615, 6.58617142680172883800545352074, 7.44008953909143964250731261287, 7.52502321592806223024002675380, 8.426766434051111891829236070031, 8.622916873325483920549799157987, 9.464499930887899250102294880192, 9.686682845696963045888210594374, 10.24049866181152910309926682308, 10.41021592605496545386155357545, 11.45570590780023874320557567955, 11.59942638354748654326905038679
