| L(s) = 1 | − 0.387·2-s − 2.91·3-s − 1.84·4-s − 5-s + 1.12·6-s + 1.49·8-s + 5.48·9-s + 0.387·10-s + 4.25·11-s + 5.38·12-s + 6.60·13-s + 2.91·15-s + 3.12·16-s − 3.12·17-s − 2.12·18-s − 5.54·19-s + 1.84·20-s − 1.65·22-s − 7.98·23-s − 4.34·24-s + 25-s − 2.55·26-s − 7.23·27-s + 29-s − 1.12·30-s − 2.94·31-s − 4.19·32-s + ⋯ |
| L(s) = 1 | − 0.274·2-s − 1.68·3-s − 0.924·4-s − 0.447·5-s + 0.460·6-s + 0.527·8-s + 1.82·9-s + 0.122·10-s + 1.28·11-s + 1.55·12-s + 1.83·13-s + 0.752·15-s + 0.780·16-s − 0.758·17-s − 0.500·18-s − 1.27·19-s + 0.413·20-s − 0.351·22-s − 1.66·23-s − 0.886·24-s + 0.200·25-s − 0.501·26-s − 1.39·27-s + 0.185·29-s − 0.206·30-s − 0.529·31-s − 0.741·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.387T + 2T^{2} \) |
| 3 | \( 1 + 2.91T + 3T^{2} \) |
| 11 | \( 1 - 4.25T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 0.404T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.61593924495467034278877271364, −6.45720409494108612655546492581, −6.30242871447782698278635218068, −5.64684704801135676405057254488, −4.54214152849836831334727137201, −4.19629985562856599311241651441, −3.66605701774550700757023638248, −1.75542971322059426231305810688, −0.936635164900566428408503180134, 0,
0.936635164900566428408503180134, 1.75542971322059426231305810688, 3.66605701774550700757023638248, 4.19629985562856599311241651441, 4.54214152849836831334727137201, 5.64684704801135676405057254488, 6.30242871447782698278635218068, 6.45720409494108612655546492581, 7.61593924495467034278877271364
