| L(s) = 1 | − 2-s + 2.13·3-s + 4-s − 0.374·5-s − 2.13·6-s − 8-s + 1.55·9-s + 0.374·10-s − 5.08·11-s + 2.13·12-s + 3.63·13-s − 0.798·15-s + 16-s + 2.81·17-s − 1.55·18-s + 0.177·19-s − 0.374·20-s + 5.08·22-s − 1.78·23-s − 2.13·24-s − 4.85·25-s − 3.63·26-s − 3.08·27-s − 8.08·29-s + 0.798·30-s + 4.16·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.167·5-s − 0.871·6-s − 0.353·8-s + 0.518·9-s + 0.118·10-s − 1.53·11-s + 0.616·12-s + 1.00·13-s − 0.206·15-s + 0.250·16-s + 0.682·17-s − 0.366·18-s + 0.0406·19-s − 0.0836·20-s + 1.08·22-s − 0.371·23-s − 0.435·24-s − 0.971·25-s − 0.712·26-s − 0.593·27-s − 1.50·29-s + 0.145·30-s + 0.748·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 0.374T + 5T^{2} \) |
| 11 | \( 1 + 5.08T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 - 0.177T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 8.08T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6.35T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 + 6.00T + 61T^{2} \) |
| 67 | \( 1 - 6.68T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 0.0930T + 97T^{2} \) |
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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
−8.204202167667610375850448772630, −7.72702960076466844315443805694, −6.94216580485337714497792649846, −5.86907723725940437613753266671, −5.22187209853606275451332823222, −3.85111965883161929279524636683, −3.29060590940741441127619409288, −2.42100746212651641855847448949, −1.60025662018945861488661378119, 0,
1.60025662018945861488661378119, 2.42100746212651641855847448949, 3.29060590940741441127619409288, 3.85111965883161929279524636683, 5.22187209853606275451332823222, 5.86907723725940437613753266671, 6.94216580485337714497792649846, 7.72702960076466844315443805694, 8.204202167667610375850448772630
