L(s) = 1 | − 2·2-s + 2·3-s − 10·4-s − 4·6-s − 20·7-s + 32·8-s − 3·9-s − 22·11-s − 20·12-s − 80·13-s + 40·14-s + 44·16-s + 124·17-s + 6·18-s + 72·19-s − 40·21-s + 44·22-s + 98·23-s + 64·24-s + 160·26-s + 34·27-s + 200·28-s + 144·29-s − 34·31-s − 248·32-s − 44·33-s − 248·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.384·3-s − 5/4·4-s − 0.272·6-s − 1.07·7-s + 1.41·8-s − 1/9·9-s − 0.603·11-s − 0.481·12-s − 1.70·13-s + 0.763·14-s + 0.687·16-s + 1.76·17-s + 0.0785·18-s + 0.869·19-s − 0.415·21-s + 0.426·22-s + 0.888·23-s + 0.544·24-s + 1.20·26-s + 0.242·27-s + 1.34·28-s + 0.922·29-s − 0.196·31-s − 1.37·32-s − 0.232·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8112800953\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8112800953\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + p T + 7 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4214 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 98 T + 22847 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 54 T + 101843 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 492 T + 348862 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 634 T + 458975 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 840 T + 528794 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 754 T + 742455 T^{2} + 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2194 T + 2966547 T^{2} + 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.70370959362126771656549210330, −11.14789295300439251960495037679, −10.28222194910455251474483762177, −9.956045329144815498649454801127, −9.878274484001140891789466548432, −9.455042437695519738736060166774, −8.672239455036832000166342020144, −8.661184254947505878776942828084, −7.79999803744748503596178972918, −7.37397139702865593209739441604, −7.19164376289760137221115796028, −6.16147922627228222629750694129, −5.35669589232422925477639914936, −5.23048352623513498688918533164, −4.44286433801095878739877423031, −3.78214299037713358390900086216, −2.96310386999313111448460743186, −2.62939246730639744897079258132, −1.09034728968805267426435460662, −0.47370028119489760899993884956,
0.47370028119489760899993884956, 1.09034728968805267426435460662, 2.62939246730639744897079258132, 2.96310386999313111448460743186, 3.78214299037713358390900086216, 4.44286433801095878739877423031, 5.23048352623513498688918533164, 5.35669589232422925477639914936, 6.16147922627228222629750694129, 7.19164376289760137221115796028, 7.37397139702865593209739441604, 7.79999803744748503596178972918, 8.661184254947505878776942828084, 8.672239455036832000166342020144, 9.455042437695519738736060166774, 9.878274484001140891789466548432, 9.956045329144815498649454801127, 10.28222194910455251474483762177, 11.14789295300439251960495037679, 11.70370959362126771656549210330