| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 11.9·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 23.8·10-s − 9.57·11-s + 12·12-s − 59.5·13-s + 14·14-s + 35.7·15-s + 16·16-s + 52.8·17-s − 18·18-s − 19·19-s + 47.7·20-s − 21·21-s + 19.1·22-s − 103.·23-s − 24·24-s + 17.3·25-s + 119.·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.06·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.754·10-s − 0.262·11-s + 0.288·12-s − 1.26·13-s + 0.267·14-s + 0.616·15-s + 0.250·16-s + 0.753·17-s − 0.235·18-s − 0.229·19-s + 0.533·20-s − 0.218·21-s + 0.185·22-s − 0.939·23-s − 0.204·24-s + 0.138·25-s + 0.897·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 11.9T + 125T^{2} \) |
| 11 | \( 1 + 9.57T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 182.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 100.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 234.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 253.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 410.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 250.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 272.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 544.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 879.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 200.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 954.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−9.612029698837025473311530061349, −8.773158267228061085172086766732, −7.75737096618784202256028070463, −7.07965234174733616014271917819, −6.00059661493869955474218710313, −5.16707838113381127767239126593, −3.63110788759666411519177057018, −2.46376941903874804906698979457, −1.71521295673885339944884725488, 0,
1.71521295673885339944884725488, 2.46376941903874804906698979457, 3.63110788759666411519177057018, 5.16707838113381127767239126593, 6.00059661493869955474218710313, 7.07965234174733616014271917819, 7.75737096618784202256028070463, 8.773158267228061085172086766732, 9.612029698837025473311530061349
