| L(s) = 1 | + 2-s − 3-s + 4-s − 2.32·5-s − 6-s + 8-s + 9-s − 2.32·10-s + 5.23·11-s − 12-s − 2·13-s + 2.32·15-s + 16-s − 0.670·17-s + 18-s − 19-s − 2.32·20-s + 5.23·22-s − 8.27·23-s − 24-s + 0.426·25-s − 2·26-s − 27-s + 0.143·29-s + 2.32·30-s + 0.569·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.04·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.736·10-s + 1.57·11-s − 0.288·12-s − 0.554·13-s + 0.601·15-s + 0.250·16-s − 0.162·17-s + 0.235·18-s − 0.229·19-s − 0.520·20-s + 1.11·22-s − 1.72·23-s − 0.204·24-s + 0.0853·25-s − 0.392·26-s − 0.192·27-s + 0.0265·29-s + 0.425·30-s + 0.102·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.32T + 5T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 0.670T + 17T^{2} \) |
| 23 | \( 1 + 8.27T + 23T^{2} \) |
| 29 | \( 1 - 0.143T + 29T^{2} \) |
| 31 | \( 1 - 0.569T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 + 9.18T + 41T^{2} \) |
| 43 | \( 1 - 2.04T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 7.96T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 3.13T + 67T^{2} \) |
| 71 | \( 1 + 9.98T + 71T^{2} \) |
| 73 | \( 1 + 5.14T + 73T^{2} \) |
| 79 | \( 1 + 9.94T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 8.33T + 97T^{2} \) |
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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
−7.52485604418989956667384650383, −7.04059725999307889672243560709, −6.26031300083983698022084171547, −5.70705546219000648520710880016, −4.67139449710739559008605840633, −4.04967597473983995984351369336, −3.70475347298016665464723402694, −2.42893912143976959394509898386, −1.34984722071411580129841777204, 0,
1.34984722071411580129841777204, 2.42893912143976959394509898386, 3.70475347298016665464723402694, 4.04967597473983995984351369336, 4.67139449710739559008605840633, 5.70705546219000648520710880016, 6.26031300083983698022084171547, 7.04059725999307889672243560709, 7.52485604418989956667384650383
