| L(s) = 1 | + 4·2-s − 2·3-s + 8·4-s + 5·5-s − 8·6-s + 32·8-s + 27·9-s + 20·10-s − 32·11-s − 16·12-s − 76·13-s − 10·15-s + 128·16-s − 26·17-s + 108·18-s − 100·19-s + 40·20-s − 128·22-s + 78·23-s − 64·24-s − 304·26-s − 154·27-s − 100·29-s − 40·30-s + 108·31-s + 256·32-s + 64·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.384·3-s + 4-s + 0.447·5-s − 0.544·6-s + 1.41·8-s + 9-s + 0.632·10-s − 0.877·11-s − 0.384·12-s − 1.62·13-s − 0.172·15-s + 2·16-s − 0.370·17-s + 1.41·18-s − 1.20·19-s + 0.447·20-s − 1.24·22-s + 0.707·23-s − 0.544·24-s − 2.29·26-s − 1.09·27-s − 0.640·29-s − 0.243·30-s + 0.625·31-s + 1.41·32-s + 0.337·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.945473872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.945473872\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 32 T - 307 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T - 4237 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 100 T + 3141 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 108 T - 18127 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 266 T + 20103 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 514 T + 160373 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 148873 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 500 T + 44621 T^{2} + 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 518 T + 41343 T^{2} - 518 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 126 T - 284887 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 878 T + 381867 T^{2} - 878 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 600 T - 133039 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 282 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 150 T - 682469 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−12.54712674832775462319107104116, −11.35096487950710421144254125816, −11.01473925262722107671989393189, −10.41520018845263075016361902192, −10.32822786445233429690194285638, −9.591548520721898857592312247374, −9.176229285482768075285099120569, −8.311955385578660239386125764245, −7.61834804201574970677668595993, −7.22348633186634121494028982610, −7.00126761520659649683611023234, −5.97991983867936418588493977696, −5.70143505802079760650968466587, −4.99240672876561247558568466609, −4.59483832212198430216952810042, −4.23264448417546051908294080439, −3.42927418173034436819367936490, −2.27394626245577469507897501515, −2.11216126027956201076185150783, −0.71771042518030613319007820161,
0.71771042518030613319007820161, 2.11216126027956201076185150783, 2.27394626245577469507897501515, 3.42927418173034436819367936490, 4.23264448417546051908294080439, 4.59483832212198430216952810042, 4.99240672876561247558568466609, 5.70143505802079760650968466587, 5.97991983867936418588493977696, 7.00126761520659649683611023234, 7.22348633186634121494028982610, 7.61834804201574970677668595993, 8.311955385578660239386125764245, 9.176229285482768075285099120569, 9.591548520721898857592312247374, 10.32822786445233429690194285638, 10.41520018845263075016361902192, 11.01473925262722107671989393189, 11.35096487950710421144254125816, 12.54712674832775462319107104116
