| L(s) = 1 | + 1.56·2-s + 0.438·4-s − 5-s + 4.56·7-s − 2.43·8-s − 1.56·10-s − 2.56·11-s + 7.12·14-s − 4.68·16-s − 2.56·17-s − 3.12·19-s − 0.438·20-s − 4·22-s + 6.56·23-s + 25-s + 1.99·28-s + 1.12·29-s − 6·31-s − 2.43·32-s − 4·34-s − 4.56·35-s − 1.68·37-s − 4.87·38-s + 2.43·40-s + 0.561·41-s − 5.12·43-s − 1.12·44-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s − 0.447·5-s + 1.72·7-s − 0.862·8-s − 0.493·10-s − 0.772·11-s + 1.90·14-s − 1.17·16-s − 0.621·17-s − 0.716·19-s − 0.0980·20-s − 0.852·22-s + 1.36·23-s + 0.200·25-s + 0.377·28-s + 0.208·29-s − 1.07·31-s − 0.431·32-s − 0.685·34-s − 0.771·35-s − 0.276·37-s − 0.791·38-s + 0.385·40-s + 0.0876·41-s − 0.781·43-s − 0.169·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 - 7.68T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 7.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.41521058703008263132812044952, −6.85979989724566966429228511734, −5.79376183270463591176874316121, −5.24562716356405514660489227274, −4.61231143778379620110568609475, −4.27061092557917247733345926989, −3.24745175263097006770272415338, −2.45980293618843288917115321302, −1.48739961317658118421891326848, 0,
1.48739961317658118421891326848, 2.45980293618843288917115321302, 3.24745175263097006770272415338, 4.27061092557917247733345926989, 4.61231143778379620110568609475, 5.24562716356405514660489227274, 5.79376183270463591176874316121, 6.85979989724566966429228511734, 7.41521058703008263132812044952
