| L(s) = 1 | + 2-s − 3-s + 4-s + 1.48·5-s − 6-s − 3.17·7-s + 8-s + 9-s + 1.48·10-s + 0.113·11-s − 12-s − 13-s − 3.17·14-s − 1.48·15-s + 16-s − 3.88·17-s + 18-s − 0.323·19-s + 1.48·20-s + 3.17·21-s + 0.113·22-s + 3.28·23-s − 24-s − 2.79·25-s − 26-s − 27-s − 3.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.663·5-s − 0.408·6-s − 1.19·7-s + 0.353·8-s + 0.333·9-s + 0.469·10-s + 0.0341·11-s − 0.288·12-s − 0.277·13-s − 0.848·14-s − 0.383·15-s + 0.250·16-s − 0.942·17-s + 0.235·18-s − 0.0742·19-s + 0.331·20-s + 0.692·21-s + 0.0241·22-s + 0.685·23-s − 0.204·24-s − 0.559·25-s − 0.196·26-s − 0.192·27-s − 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 - 0.113T + 11T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 19 | \( 1 + 0.323T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.13T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 + 8.63T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 - 8.30T + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−7.26706408998311539724824146934, −6.43603212848696592540328396929, −6.15548572962393475282276050311, −5.56538501151071242585834729391, −4.58508531294142276728209642005, −4.12950734362984272219619659588, −2.99747411698235781414868507446, −2.48440293351693844847980025565, −1.33413944209964897072141055424, 0,
1.33413944209964897072141055424, 2.48440293351693844847980025565, 2.99747411698235781414868507446, 4.12950734362984272219619659588, 4.58508531294142276728209642005, 5.56538501151071242585834729391, 6.15548572962393475282276050311, 6.43603212848696592540328396929, 7.26706408998311539724824146934
