| L(s) = 1 | − 0.326·2-s − 3.04·3-s − 7.89·4-s + 0.994·6-s − 25.8·7-s + 5.18·8-s − 17.7·9-s − 35.5·11-s + 24.0·12-s − 89.5·13-s + 8.42·14-s + 61.4·16-s + 82.5·17-s + 5.77·18-s + 123.·19-s + 78.7·21-s + 11.5·22-s − 15.0·23-s − 15.7·24-s + 29.2·26-s + 136.·27-s + 203.·28-s + 70.3·29-s − 4.21·31-s − 61.5·32-s + 108.·33-s − 26.9·34-s + ⋯ |
| L(s) = 1 | − 0.115·2-s − 0.586·3-s − 0.986·4-s + 0.0676·6-s − 1.39·7-s + 0.229·8-s − 0.656·9-s − 0.974·11-s + 0.578·12-s − 1.91·13-s + 0.160·14-s + 0.960·16-s + 1.17·17-s + 0.0756·18-s + 1.48·19-s + 0.818·21-s + 0.112·22-s − 0.136·23-s − 0.134·24-s + 0.220·26-s + 0.971·27-s + 1.37·28-s + 0.450·29-s − 0.0243·31-s − 0.339·32-s + 0.571·33-s − 0.135·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 71 | \( 1 - 71T \) |
| good | 2 | \( 1 + 0.326T + 8T^{2} \) |
| 3 | \( 1 + 3.04T + 27T^{2} \) |
| 7 | \( 1 + 25.8T + 343T^{2} \) |
| 11 | \( 1 + 35.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 70.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.21T + 2.97e4T^{2} \) |
| 37 | \( 1 - 41.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 87.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 715.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 705.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 341.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 945.T + 3.00e5T^{2} \) |
| 73 | \( 1 + 782.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 314.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.45e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 272.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 495.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.629017590020507652636340228492, −7.64807633729143200896899749762, −7.10324385251643136537436790595, −5.74342500702530713816299985733, −5.45188459136612555171322455139, −4.59419479567893632822972739515, −3.28506146669837419409853935785, −2.72353826662584099374847509605, −0.73958096837549760244965839496, 0,
0.73958096837549760244965839496, 2.72353826662584099374847509605, 3.28506146669837419409853935785, 4.59419479567893632822972739515, 5.45188459136612555171322455139, 5.74342500702530713816299985733, 7.10324385251643136537436790595, 7.64807633729143200896899749762, 8.629017590020507652636340228492
