| L(s) = 1 | + 0.810·2-s + 3-s − 1.34·4-s − 3.50·5-s + 0.810·6-s − 0.645·7-s − 2.70·8-s + 9-s − 2.84·10-s + 1.66·11-s − 1.34·12-s + 13-s − 0.522·14-s − 3.50·15-s + 0.493·16-s + 4.69·17-s + 0.810·18-s + 3.69·19-s + 4.71·20-s − 0.645·21-s + 1.34·22-s + 0.624·23-s − 2.70·24-s + 7.30·25-s + 0.810·26-s + 27-s + 0.866·28-s + ⋯ |
| L(s) = 1 | + 0.572·2-s + 0.577·3-s − 0.671·4-s − 1.56·5-s + 0.330·6-s − 0.243·7-s − 0.957·8-s + 0.333·9-s − 0.898·10-s + 0.502·11-s − 0.387·12-s + 0.277·13-s − 0.139·14-s − 0.905·15-s + 0.123·16-s + 1.13·17-s + 0.190·18-s + 0.848·19-s + 1.05·20-s − 0.140·21-s + 0.287·22-s + 0.130·23-s − 0.552·24-s + 1.46·25-s + 0.158·26-s + 0.192·27-s + 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.810T + 2T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 0.645T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 23 | \( 1 - 0.624T + 23T^{2} \) |
| 29 | \( 1 + 2.00T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 6.13T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 0.00767T + 47T^{2} \) |
| 53 | \( 1 + 0.698T + 53T^{2} \) |
| 59 | \( 1 + 0.481T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 3.15T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.056435642136592709355936354805, −7.50377937709498069331734844557, −6.71774143748406011355952360099, −5.65710698685515671897205809263, −4.88282318730310900909074431306, −4.08343080883039381663726613753, −3.43506411932955574473244517024, −3.13150385659857666181730640027, −1.31157722073191894421831225248, 0,
1.31157722073191894421831225248, 3.13150385659857666181730640027, 3.43506411932955574473244517024, 4.08343080883039381663726613753, 4.88282318730310900909074431306, 5.65710698685515671897205809263, 6.71774143748406011355952360099, 7.50377937709498069331734844557, 8.056435642136592709355936354805
