| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s + 2·13-s + 4·14-s + 16-s − 17-s − 18-s + 4·19-s − 8·21-s − 6·22-s − 2·24-s − 5·25-s − 2·26-s − 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.74·21-s − 1.27·22-s − 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 53 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.94490645430136, −13.69852899083780, −13.22231979959053, −12.55025888240168, −12.02154445029476, −11.62320482388516, −11.14809735164956, −10.26915863766195, −9.913690023426038, −9.412169969260424, −9.164341718641459, −8.657517705598510, −8.288073863854331, −7.448427892571457, −7.136903309564854, −6.517506794893062, −6.079409750598451, −5.641274504165794, −4.497307505005756, −3.803208022523111, −3.496778701906167, −3.019415377639852, −2.332429983945447, −1.606189332003256, −0.9612081655631656, 0,
0.9612081655631656, 1.606189332003256, 2.332429983945447, 3.019415377639852, 3.496778701906167, 3.803208022523111, 4.497307505005756, 5.641274504165794, 6.079409750598451, 6.517506794893062, 7.136903309564854, 7.448427892571457, 8.288073863854331, 8.657517705598510, 9.164341718641459, 9.412169969260424, 9.913690023426038, 10.26915863766195, 11.14809735164956, 11.62320482388516, 12.02154445029476, 12.55025888240168, 13.22231979959053, 13.69852899083780, 13.94490645430136
