| L(s) = 1 | − 0.539·3-s − 5-s + 7-s − 2.70·9-s − 11-s − 0.539·13-s + 0.539·15-s + 7.21·17-s + 6.49·19-s − 0.539·21-s − 8.38·23-s + 25-s + 3.07·27-s − 9.41·29-s + 3.51·31-s + 0.539·33-s − 35-s − 10.2·37-s + 0.290·39-s − 8.43·41-s + 10.7·43-s + 2.70·45-s + 4.29·47-s + 49-s − 3.89·51-s − 4.29·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.311·3-s − 0.447·5-s + 0.377·7-s − 0.903·9-s − 0.301·11-s − 0.149·13-s + 0.139·15-s + 1.75·17-s + 1.49·19-s − 0.117·21-s − 1.74·23-s + 0.200·25-s + 0.592·27-s − 1.74·29-s + 0.630·31-s + 0.0938·33-s − 0.169·35-s − 1.68·37-s + 0.0465·39-s − 1.31·41-s + 1.64·43-s + 0.403·45-s + 0.626·47-s + 0.142·49-s − 0.545·51-s − 0.589·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 13 | \( 1 + 0.539T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 8.38T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 8.43T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 7.94T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.67508911620405833013336352278, −7.24548500288981887177241931111, −6.08326234360430668532135580951, −5.45542537274277665947386298506, −5.12762405532773557499795430321, −3.85094694570728734573713939845, −3.36862988149174000402508231741, −2.34856517678035047256543758287, −1.19853496852148766722023279708, 0,
1.19853496852148766722023279708, 2.34856517678035047256543758287, 3.36862988149174000402508231741, 3.85094694570728734573713939845, 5.12762405532773557499795430321, 5.45542537274277665947386298506, 6.08326234360430668532135580951, 7.24548500288981887177241931111, 7.67508911620405833013336352278
