| L(s) = 1 | + 0.0187·2-s + 1.61·3-s − 1.99·4-s − 1.12·5-s + 0.0303·6-s − 0.878·7-s − 0.0751·8-s − 0.386·9-s − 0.0211·10-s + 5.47·11-s − 3.23·12-s − 5.27·13-s − 0.0164·14-s − 1.82·15-s + 3.99·16-s + 17-s − 0.00725·18-s − 3.24·19-s + 2.25·20-s − 1.41·21-s + 0.102·22-s − 0.753·23-s − 0.121·24-s − 3.72·25-s − 0.0990·26-s − 5.47·27-s + 1.75·28-s + ⋯ |
| L(s) = 1 | + 0.0132·2-s + 0.933·3-s − 0.999·4-s − 0.504·5-s + 0.0123·6-s − 0.331·7-s − 0.0265·8-s − 0.128·9-s − 0.00669·10-s + 1.65·11-s − 0.933·12-s − 1.46·13-s − 0.00440·14-s − 0.470·15-s + 0.999·16-s + 0.242·17-s − 0.00170·18-s − 0.744·19-s + 0.503·20-s − 0.309·21-s + 0.0219·22-s − 0.157·23-s − 0.0247·24-s − 0.745·25-s − 0.0194·26-s − 1.05·27-s + 0.331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0187T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 0.878T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 + 0.753T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 0.928T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.803T + 47T^{2} \) |
| 53 | \( 1 - 1.02T + 53T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.56T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 + 5.86T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−9.416949896984192884181078734728, −8.866455386312379252444403900475, −8.007683839521449358481866402181, −7.29416288070936910577182919862, −6.13251175494654766095401474842, −4.99003404781843488160413140889, −3.91586395501902480603224491698, −3.46396066755923937626012595197, −1.97483335730951043689896294454, 0,
1.97483335730951043689896294454, 3.46396066755923937626012595197, 3.91586395501902480603224491698, 4.99003404781843488160413140889, 6.13251175494654766095401474842, 7.29416288070936910577182919862, 8.007683839521449358481866402181, 8.866455386312379252444403900475, 9.416949896984192884181078734728
