| L(s) = 1 | − 2·2-s + 8.30·3-s + 4·4-s − 16.6·6-s + 11.9·7-s − 8·8-s + 41.9·9-s + 54.3·11-s + 33.2·12-s + 67.8·13-s − 23.9·14-s + 16·16-s + 10.6·17-s − 83.9·18-s + 22.9·19-s + 99.3·21-s − 108.·22-s + 147.·23-s − 66.4·24-s − 135.·26-s + 124.·27-s + 47.8·28-s + 205.·29-s − 87.6·31-s − 32·32-s + 451.·33-s − 21.2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.59·3-s + 0.5·4-s − 1.13·6-s + 0.645·7-s − 0.353·8-s + 1.55·9-s + 1.48·11-s + 0.799·12-s + 1.44·13-s − 0.456·14-s + 0.250·16-s + 0.151·17-s − 1.09·18-s + 0.277·19-s + 1.03·21-s − 1.05·22-s + 1.33·23-s − 0.565·24-s − 1.02·26-s + 0.886·27-s + 0.322·28-s + 1.31·29-s − 0.507·31-s − 0.176·32-s + 2.38·33-s − 0.107·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.933968579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.933968579\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8.30T + 27T^{2} \) |
| 7 | \( 1 - 11.9T + 343T^{2} \) |
| 11 | \( 1 - 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 413.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 67.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 66.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 101.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 405.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 477.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 857.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 549.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 39.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−9.035498750786148777313786603501, −8.610055840474365079302271785076, −8.078936411383808078814557644818, −7.03963551977920677093940365914, −6.41642167248270792478702351879, −4.92139498869641512839693129178, −3.68033320470698989089948981847, −3.17331508518770915275983494402, −1.74736175494083124537429285420, −1.21844710552070063501257751982,
1.21844710552070063501257751982, 1.74736175494083124537429285420, 3.17331508518770915275983494402, 3.68033320470698989089948981847, 4.92139498869641512839693129178, 6.41642167248270792478702351879, 7.03963551977920677093940365914, 8.078936411383808078814557644818, 8.610055840474365079302271785076, 9.035498750786148777313786603501
