L(s) = 1 | + 2-s + 3-s + 4-s − 0.198·5-s + 6-s − 5.04·7-s + 8-s + 9-s − 0.198·10-s + 11-s + 12-s − 1.64·13-s − 5.04·14-s − 0.198·15-s + 16-s + 3.49·17-s + 18-s + 2.02·19-s − 0.198·20-s − 5.04·21-s + 22-s − 2.46·23-s + 24-s − 4.96·25-s − 1.64·26-s + 27-s − 5.04·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0885·5-s + 0.408·6-s − 1.90·7-s + 0.353·8-s + 0.333·9-s − 0.0626·10-s + 0.301·11-s + 0.288·12-s − 0.455·13-s − 1.34·14-s − 0.0511·15-s + 0.250·16-s + 0.847·17-s + 0.235·18-s + 0.465·19-s − 0.0442·20-s − 1.10·21-s + 0.213·22-s − 0.514·23-s + 0.204·24-s − 0.992·25-s − 0.322·26-s + 0.192·27-s − 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 + 0.198T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 0.643T + 37T^{2} \) |
| 41 | \( 1 + 9.59T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 67 | \( 1 - 2.26T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 - 1.34T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.76285434182091031213023525233, −7.38238316482283550550605590467, −6.48215195975048043926902574713, −5.94919512395427848792639174605, −5.10719116081356836072120451927, −3.89075503054427967814103640628, −3.49986036789977186627125003681, −2.80696220707996981693811692713, −1.71301326040849372311966802688, 0,
1.71301326040849372311966802688, 2.80696220707996981693811692713, 3.49986036789977186627125003681, 3.89075503054427967814103640628, 5.10719116081356836072120451927, 5.94919512395427848792639174605, 6.48215195975048043926902574713, 7.38238316482283550550605590467, 7.76285434182091031213023525233