| L(s) = 1 | − 0.101·2-s − 0.616·3-s − 1.98·4-s − 5-s + 0.0626·6-s + 0.405·8-s − 2.62·9-s + 0.101·10-s − 5.17·11-s + 1.22·12-s + 0.378·13-s + 0.616·15-s + 3.93·16-s − 3.17·17-s + 0.266·18-s − 3.27·19-s + 1.98·20-s + 0.526·22-s − 3.27·23-s − 0.249·24-s + 25-s − 0.0385·26-s + 3.46·27-s + 7.70·29-s − 0.0626·30-s + 5.95·31-s − 1.21·32-s + ⋯ |
| L(s) = 1 | − 0.0719·2-s − 0.355·3-s − 0.994·4-s − 0.447·5-s + 0.0255·6-s + 0.143·8-s − 0.873·9-s + 0.0321·10-s − 1.56·11-s + 0.353·12-s + 0.105·13-s + 0.159·15-s + 0.984·16-s − 0.771·17-s + 0.0628·18-s − 0.752·19-s + 0.444·20-s + 0.112·22-s − 0.682·23-s − 0.0510·24-s + 0.200·25-s − 0.00755·26-s + 0.666·27-s + 1.43·29-s − 0.0114·30-s + 1.06·31-s − 0.214·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.101T + 2T^{2} \) |
| 3 | \( 1 + 0.616T + 3T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 - 0.378T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 7.42T + 43T^{2} \) |
| 47 | \( 1 - 7.52T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 + 0.553T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.67051893715872717398930716041, −6.59185881849397809126489205900, −5.91231997953036376611554934634, −5.28387162658121181277033647171, −4.57030821149036655894491298735, −4.10236218550449220932029526706, −2.95681325512472329762029173927, −2.40530180837448859358484871728, −0.793907138972722648881450422923, 0,
0.793907138972722648881450422923, 2.40530180837448859358484871728, 2.95681325512472329762029173927, 4.10236218550449220932029526706, 4.57030821149036655894491298735, 5.28387162658121181277033647171, 5.91231997953036376611554934634, 6.59185881849397809126489205900, 7.67051893715872717398930716041
