| L(s) = 1 | + 2.38·2-s − 3-s + 3.66·4-s − 0.777·5-s − 2.38·6-s + 0.635·7-s + 3.96·8-s + 9-s − 1.84·10-s − 5.53·11-s − 3.66·12-s − 13-s + 1.51·14-s + 0.777·15-s + 2.10·16-s + 4.48·17-s + 2.38·18-s − 4.39·19-s − 2.84·20-s − 0.635·21-s − 13.1·22-s − 0.483·23-s − 3.96·24-s − 4.39·25-s − 2.38·26-s − 27-s + 2.32·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.347·5-s − 0.971·6-s + 0.240·7-s + 1.40·8-s + 0.333·9-s − 0.584·10-s − 1.66·11-s − 1.05·12-s − 0.277·13-s + 0.404·14-s + 0.200·15-s + 0.525·16-s + 1.08·17-s + 0.560·18-s − 1.00·19-s − 0.636·20-s − 0.138·21-s − 2.81·22-s − 0.100·23-s − 0.808·24-s − 0.879·25-s − 0.466·26-s − 0.192·27-s + 0.440·28-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 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 + 0.777T + 5T^{2} \) |
| 7 | \( 1 - 0.635T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 + 0.483T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 9.84T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 + 5.52T + 79T^{2} \) |
| 83 | \( 1 + 3.60T + 83T^{2} \) |
| 89 | \( 1 + 1.32T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.927916742066260165586726159659, −7.05237574927308025292656405181, −6.38686775907942450112349492251, −5.52659631582549251330851376146, −5.10879366156758459883668378532, −4.46842294790809357481448962645, −3.56071685523924044198507734527, −2.78565408006886494030316123430, −1.82657366828000237497455743388, 0,
1.82657366828000237497455743388, 2.78565408006886494030316123430, 3.56071685523924044198507734527, 4.46842294790809357481448962645, 5.10879366156758459883668378532, 5.52659631582549251330851376146, 6.38686775907942450112349492251, 7.05237574927308025292656405181, 7.927916742066260165586726159659
