| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 0.328·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 0.657·10-s − 40.5·11-s − 12·12-s + 0.612·13-s − 14·14-s − 0.986·15-s + 16·16-s + 96.1·17-s − 18·18-s − 19·19-s + 1.31·20-s − 21·21-s + 81.1·22-s + 120.·23-s + 24·24-s − 124.·25-s − 1.22·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0294·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.0207·10-s − 1.11·11-s − 0.288·12-s + 0.0130·13-s − 0.267·14-s − 0.0169·15-s + 0.250·16-s + 1.37·17-s − 0.235·18-s − 0.229·19-s + 0.0147·20-s − 0.218·21-s + 0.786·22-s + 1.09·23-s + 0.204·24-s − 0.999·25-s − 0.00923·26-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 | \( 1 - 0.328T + 125T^{2} \) |
| 11 | \( 1 + 40.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.612T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 371.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 574.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 459.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 553.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 551.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 292.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 452.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 142.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 211.T + 9.12e5T^{2} \) |
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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
−9.534174847312183453929779768709, −8.613553034386453373820478312341, −7.61236987762489807414405843836, −7.18213803945158127101176780907, −5.72612667435928334306479931478, −5.34292277521336974653230059668, −3.88580371479474081974042565162, −2.54943555537727701932752098552, −1.27897408741936757286920054254, 0,
1.27897408741936757286920054254, 2.54943555537727701932752098552, 3.88580371479474081974042565162, 5.34292277521336974653230059668, 5.72612667435928334306479931478, 7.18213803945158127101176780907, 7.61236987762489807414405843836, 8.613553034386453373820478312341, 9.534174847312183453929779768709
