| L(s) = 1 | + 2·2-s − 5·5-s + 4·7-s − 8·8-s − 10·10-s − 48·11-s − 2·13-s + 8·14-s − 16·16-s + 228·17-s + 280·19-s − 96·22-s + 72·23-s − 4·26-s + 210·29-s − 272·31-s + 456·34-s − 20·35-s − 668·37-s + 560·38-s + 40·40-s − 198·41-s + 268·43-s + 144·46-s + 216·47-s + 343·49-s + 156·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 0.215·7-s − 0.353·8-s − 0.316·10-s − 1.31·11-s − 0.0426·13-s + 0.152·14-s − 1/4·16-s + 3.25·17-s + 3.38·19-s − 0.930·22-s + 0.652·23-s − 0.0301·26-s + 1.34·29-s − 1.57·31-s + 2.30·34-s − 0.0965·35-s − 2.96·37-s + 2.39·38-s + 0.158·40-s − 0.754·41-s + 0.950·43-s + 0.461·46-s + 0.670·47-s + 49-s + 0.404·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.870273790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.870273790\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 2193 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 210 T + 19711 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 272 T + 44193 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 334 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 198 T - 29717 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 268 T - 7683 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 216 T - 57167 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 240 T - 147779 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 302 T - 135777 T^{2} + 302 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 478 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 640 T - 83439 T^{2} - 640 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 348 T - 450683 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1534 T + 1440483 T^{2} - 1534 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.15654237602487405573337490213, −9.880599521252069768472748533219, −9.042345667320217032631065944274, −8.994855848758337873062899141920, −8.224241849301056216918337304864, −7.69031941439357464882675871816, −7.49682792104555461771904144127, −7.34696231272868379192233176262, −6.66862086101920496331719031501, −5.67861809605018058164066081574, −5.49414733823382487935011348593, −5.15419101051553368630762217472, −5.12196319973062219200578936599, −4.07213867728197129802241045458, −3.45053147291626498129051716096, −3.11130975132747923062393873450, −3.02986739073575827442992807115, −1.83626307391308889629290670794, −1.03960538447744075143387245361, −0.64242935669352938397950764797,
0.64242935669352938397950764797, 1.03960538447744075143387245361, 1.83626307391308889629290670794, 3.02986739073575827442992807115, 3.11130975132747923062393873450, 3.45053147291626498129051716096, 4.07213867728197129802241045458, 5.12196319973062219200578936599, 5.15419101051553368630762217472, 5.49414733823382487935011348593, 5.67861809605018058164066081574, 6.66862086101920496331719031501, 7.34696231272868379192233176262, 7.49682792104555461771904144127, 7.69031941439357464882675871816, 8.224241849301056216918337304864, 8.994855848758337873062899141920, 9.042345667320217032631065944274, 9.880599521252069768472748533219, 10.15654237602487405573337490213
