| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s − 5·11-s − 2·12-s + 2·14-s − 4·16-s − 2·17-s − 4·18-s − 6·19-s − 21-s − 10·22-s + 6·23-s + 5·27-s + 2·28-s + 2·29-s − 8·31-s − 8·32-s + 5·33-s − 4·34-s − 4·36-s + 10·37-s − 12·38-s − 2·42-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.577·12-s + 0.534·14-s − 16-s − 0.485·17-s − 0.942·18-s − 1.37·19-s − 0.218·21-s − 2.13·22-s + 1.25·23-s + 0.962·27-s + 0.377·28-s + 0.371·29-s − 1.43·31-s − 1.41·32-s + 0.870·33-s − 0.685·34-s − 2/3·36-s + 1.64·37-s − 1.94·38-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321878847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321878847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.87713593707457, −14.72342517295282, −14.18734802954449, −13.21161607081113, −13.03695777099817, −12.90744687695498, −11.92849744970786, −11.53674108530119, −11.00322075776664, −10.68872011793016, −9.987220026230260, −9.036909698622827, −8.593765966130535, −7.991264745569988, −7.207227467384355, −6.585417257315815, −6.039698519805357, −5.492185315614515, −4.935866654265001, −4.655804758261884, −3.839459188943872, −2.953634124435586, −2.622034913265875, −1.766561794264613, −0.3364041879832534,
0.3364041879832534, 1.766561794264613, 2.622034913265875, 2.953634124435586, 3.839459188943872, 4.655804758261884, 4.935866654265001, 5.492185315614515, 6.039698519805357, 6.585417257315815, 7.207227467384355, 7.991264745569988, 8.593765966130535, 9.036909698622827, 9.987220026230260, 10.68872011793016, 11.00322075776664, 11.53674108530119, 11.92849744970786, 12.90744687695498, 13.03695777099817, 13.21161607081113, 14.18734802954449, 14.72342517295282, 14.87713593707457
