| L(s) = 1 | + 2-s + 2.32·3-s + 4-s − 0.739·5-s + 2.32·6-s − 4.35·7-s + 8-s + 2.41·9-s − 0.739·10-s − 3.39·11-s + 2.32·12-s − 4.11·13-s − 4.35·14-s − 1.72·15-s + 16-s − 4.99·17-s + 2.41·18-s + 6.01·19-s − 0.739·20-s − 10.1·21-s − 3.39·22-s − 0.384·23-s + 2.32·24-s − 4.45·25-s − 4.11·26-s − 1.35·27-s − 4.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 0.5·4-s − 0.330·5-s + 0.950·6-s − 1.64·7-s + 0.353·8-s + 0.805·9-s − 0.233·10-s − 1.02·11-s + 0.671·12-s − 1.14·13-s − 1.16·14-s − 0.444·15-s + 0.250·16-s − 1.21·17-s + 0.569·18-s + 1.38·19-s − 0.165·20-s − 2.21·21-s − 0.723·22-s − 0.0801·23-s + 0.475·24-s − 0.890·25-s − 0.807·26-s − 0.261·27-s − 0.823·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 0.739T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 + 4.99T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 + 0.384T + 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 - 0.295T + 31T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 - 6.14T + 61T^{2} \) |
| 67 | \( 1 + 16.2T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 + 7.99T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−8.426302764818817505227657461051, −7.53808120466511994034495117265, −7.09426381336559881136386965853, −6.20083186282114229958834571268, −5.20388551287679251490242157919, −4.30495993981388071686445334593, −3.28319258631204152208331169248, −2.94627120592197926427816270577, −2.14825687608369699855321346608, 0,
2.14825687608369699855321346608, 2.94627120592197926427816270577, 3.28319258631204152208331169248, 4.30495993981388071686445334593, 5.20388551287679251490242157919, 6.20083186282114229958834571268, 7.09426381336559881136386965853, 7.53808120466511994034495117265, 8.426302764818817505227657461051
