| L(s) = 1 | − 2.15·2-s − 1.38·3-s + 2.64·4-s + 2.98·6-s + 2.62·7-s − 1.38·8-s − 1.08·9-s − 1.64·11-s − 3.65·12-s − 2.44·13-s − 5.65·14-s − 2.29·16-s + 0.578·17-s + 2.34·18-s + 5.20·19-s − 3.63·21-s + 3.54·22-s − 8.22·23-s + 1.92·24-s + 5.26·26-s + 5.65·27-s + 6.94·28-s + 0.766·29-s + 4.21·31-s + 7.72·32-s + 2.27·33-s − 1.24·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.798·3-s + 1.32·4-s + 1.21·6-s + 0.992·7-s − 0.491·8-s − 0.362·9-s − 0.495·11-s − 1.05·12-s − 0.677·13-s − 1.51·14-s − 0.573·16-s + 0.140·17-s + 0.552·18-s + 1.19·19-s − 0.792·21-s + 0.755·22-s − 1.71·23-s + 0.392·24-s + 1.03·26-s + 1.08·27-s + 1.31·28-s + 0.142·29-s + 0.756·31-s + 1.36·32-s + 0.395·33-s − 0.213·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 0.578T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 - 0.766T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 41 | \( 1 + 1.64T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 8.85T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.743341749785100834165852255288, −8.812532070235157518172524099048, −7.906873858887505922332839729584, −7.57158604200885108249162812637, −6.35516781159197621411732743103, −5.40184522088152370175664060989, −4.52858117071984156424919149680, −2.67472831736338453149198563234, −1.40876692429763872414559496937, 0,
1.40876692429763872414559496937, 2.67472831736338453149198563234, 4.52858117071984156424919149680, 5.40184522088152370175664060989, 6.35516781159197621411732743103, 7.57158604200885108249162812637, 7.906873858887505922332839729584, 8.812532070235157518172524099048, 9.743341749785100834165852255288
