| L(s) = 1 | − 0.311·2-s − 1.90·4-s − 5-s − 0.903·7-s + 1.21·8-s + 0.311·10-s + 1.52·11-s − 0.622·13-s + 0.280·14-s + 3.42·16-s + 7.95·17-s − 1.09·19-s + 1.90·20-s − 0.474·22-s − 7.52·23-s + 25-s + 0.193·26-s + 1.71·28-s + 29-s − 6.90·31-s − 3.49·32-s − 2.47·34-s + 0.903·35-s + 3.95·37-s + 0.341·38-s − 1.21·40-s − 3.67·41-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.951·4-s − 0.447·5-s − 0.341·7-s + 0.429·8-s + 0.0983·10-s + 0.459·11-s − 0.172·13-s + 0.0750·14-s + 0.857·16-s + 1.92·17-s − 0.251·19-s + 0.425·20-s − 0.101·22-s − 1.56·23-s + 0.200·25-s + 0.0379·26-s + 0.324·28-s + 0.185·29-s − 1.23·31-s − 0.617·32-s − 0.424·34-s + 0.152·35-s + 0.650·37-s + 0.0553·38-s − 0.192·40-s − 0.573·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−9.485766913754565268795126136905, −8.167293485534335815797443050122, −8.027986117108595340978255802960, −6.84901370512058031546813207168, −5.80385595673914163595518467365, −4.97634793852087336703707632075, −3.91819856904807039536980891951, −3.30021799707193155491695692432, −1.50019452928248440997371526910, 0,
1.50019452928248440997371526910, 3.30021799707193155491695692432, 3.91819856904807039536980891951, 4.97634793852087336703707632075, 5.80385595673914163595518467365, 6.84901370512058031546813207168, 8.027986117108595340978255802960, 8.167293485534335815797443050122, 9.485766913754565268795126136905
