| L(s) = 1 | − 2-s + 1.49·3-s + 4-s + 2.18·5-s − 1.49·6-s − 3.55·7-s − 8-s − 0.757·9-s − 2.18·10-s − 11-s + 1.49·12-s − 0.628·13-s + 3.55·14-s + 3.27·15-s + 16-s − 0.874·17-s + 0.757·18-s + 6.85·19-s + 2.18·20-s − 5.32·21-s + 22-s − 0.146·23-s − 1.49·24-s − 0.214·25-s + 0.628·26-s − 5.62·27-s − 3.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.864·3-s + 0.5·4-s + 0.978·5-s − 0.611·6-s − 1.34·7-s − 0.353·8-s − 0.252·9-s − 0.691·10-s − 0.301·11-s + 0.432·12-s − 0.174·13-s + 0.950·14-s + 0.845·15-s + 0.250·16-s − 0.212·17-s + 0.178·18-s + 1.57·19-s + 0.489·20-s − 1.16·21-s + 0.213·22-s − 0.0306·23-s − 0.305·24-s − 0.0428·25-s + 0.123·26-s − 1.08·27-s − 0.671·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 13 | \( 1 + 0.628T + 13T^{2} \) |
| 17 | \( 1 + 0.874T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 0.146T + 23T^{2} \) |
| 29 | \( 1 + 0.352T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 - 0.816T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + 4.37T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 + 3.29T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.67T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.136453520420526197224652329143, −7.32713314840041332844328474074, −6.74745510006178355544000789426, −5.80185968060202461667025561899, −5.40861829510021745689583121041, −3.86571312195261668820389179274, −3.01272399440183712060002343354, −2.56253412262881556826389088370, −1.50485969669849412634337702389, 0,
1.50485969669849412634337702389, 2.56253412262881556826389088370, 3.01272399440183712060002343354, 3.86571312195261668820389179274, 5.40861829510021745689583121041, 5.80185968060202461667025561899, 6.74745510006178355544000789426, 7.32713314840041332844328474074, 8.136453520420526197224652329143
