| L(s) = 1 | + 0.335·2-s − 1.29·3-s − 7.88·4-s − 9.09·5-s − 0.432·6-s + 17.7·7-s − 5.32·8-s − 25.3·9-s − 3.04·10-s + 43.8·11-s + 10.1·12-s + 37.0·13-s + 5.93·14-s + 11.7·15-s + 61.3·16-s + 4.00·17-s − 8.48·18-s + 64.6·19-s + 71.7·20-s − 22.8·21-s + 14.6·22-s − 164.·23-s + 6.87·24-s − 42.2·25-s + 12.4·26-s + 67.5·27-s − 139.·28-s + ⋯ |
| L(s) = 1 | + 0.118·2-s − 0.248·3-s − 0.985·4-s − 0.813·5-s − 0.0294·6-s + 0.956·7-s − 0.235·8-s − 0.938·9-s − 0.0963·10-s + 1.20·11-s + 0.245·12-s + 0.790·13-s + 0.113·14-s + 0.202·15-s + 0.958·16-s + 0.0571·17-s − 0.111·18-s + 0.780·19-s + 0.801·20-s − 0.237·21-s + 0.142·22-s − 1.48·23-s + 0.0584·24-s − 0.338·25-s + 0.0937·26-s + 0.481·27-s − 0.942·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.241269203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241269203\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 0.335T + 8T^{2} \) |
| 3 | \( 1 + 1.29T + 27T^{2} \) |
| 5 | \( 1 + 9.09T + 125T^{2} \) |
| 7 | \( 1 - 17.7T + 343T^{2} \) |
| 11 | \( 1 - 43.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.00T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 95.4T + 6.89e4T^{2} \) |
| 47 | \( 1 + 180.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 166.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 651.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 47.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 395.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 423.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 374.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.776347093757957088547258107872, −8.145303396636943697769870817366, −7.62189812510397530828753604533, −6.24705546069708095251572200273, −5.66421320166503107432873498970, −4.66136827480905538668951228526, −3.99915085238535641865476897855, −3.28665868218121303777032265424, −1.61756707210605473147521636753, −0.54860862270320397929105113870,
0.54860862270320397929105113870, 1.61756707210605473147521636753, 3.28665868218121303777032265424, 3.99915085238535641865476897855, 4.66136827480905538668951228526, 5.66421320166503107432873498970, 6.24705546069708095251572200273, 7.62189812510397530828753604533, 8.145303396636943697769870817366, 8.776347093757957088547258107872
