| L(s) = 1 | + 2-s + 8·4-s + 5·5-s + 24·7-s + 23·8-s + 5·10-s + 52·11-s − 22·13-s + 24·14-s + 23·16-s + 28·17-s − 40·19-s + 40·20-s + 52·22-s − 168·23-s − 22·26-s + 192·28-s + 230·29-s + 288·31-s + 184·32-s + 28·34-s + 120·35-s − 68·37-s − 40·38-s + 115·40-s + 122·41-s + 188·43-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 4-s + 0.447·5-s + 1.29·7-s + 1.01·8-s + 0.158·10-s + 1.42·11-s − 0.469·13-s + 0.458·14-s + 0.359·16-s + 0.399·17-s − 0.482·19-s + 0.447·20-s + 0.503·22-s − 1.52·23-s − 0.165·26-s + 1.29·28-s + 1.47·29-s + 1.66·31-s + 1.01·32-s + 0.141·34-s + 0.579·35-s − 0.302·37-s − 0.170·38-s + 0.454·40-s + 0.464·41-s + 0.666·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.629834922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.629834922\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 233 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 52 T + 1373 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 22 T - 1713 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 168 T + 16057 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 230 T + 28511 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 288 T + 53153 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 122 T - 54037 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 188 T - 44163 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 256 T - 38287 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 338 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 100 T - 195379 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 742 T + 323583 T^{2} + 742 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 84 T - 293707 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 328 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 240 T - 435439 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1212 T + 897157 T^{2} - 1212 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 330 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 866 T - 162717 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.03685191165815487576769240091, −10.65493064884242599161639351004, −10.32232539591604636642634425035, −9.788485091838402497646613926602, −9.330306917979044149728506267848, −8.651110492577223436203193731980, −8.105584672680776701064768573066, −7.981751833576994079897342614293, −7.06183772594267553291840387247, −6.99534826810355526992144597728, −6.13550299264119305668137282702, −6.05969318382506731939254356349, −5.22316394867006171987224045607, −4.52249342866656905974037308526, −4.34906766025708677513403836662, −3.65351362625937442271321541951, −2.51622618802416146559391600427, −2.28047480859340653055196837158, −1.46225288388901041440202629709, −0.953746729995841975647049243499,
0.953746729995841975647049243499, 1.46225288388901041440202629709, 2.28047480859340653055196837158, 2.51622618802416146559391600427, 3.65351362625937442271321541951, 4.34906766025708677513403836662, 4.52249342866656905974037308526, 5.22316394867006171987224045607, 6.05969318382506731939254356349, 6.13550299264119305668137282702, 6.99534826810355526992144597728, 7.06183772594267553291840387247, 7.981751833576994079897342614293, 8.105584672680776701064768573066, 8.651110492577223436203193731980, 9.330306917979044149728506267848, 9.788485091838402497646613926602, 10.32232539591604636642634425035, 10.65493064884242599161639351004, 11.03685191165815487576769240091
