| L(s) = 1 | − 2-s − 3.26·3-s + 4-s − 1.57·5-s + 3.26·6-s + 7-s − 8-s + 7.63·9-s + 1.57·10-s + 3.36·11-s − 3.26·12-s + 4.38·13-s − 14-s + 5.12·15-s + 16-s + 6.97·17-s − 7.63·18-s + 1.24·19-s − 1.57·20-s − 3.26·21-s − 3.36·22-s − 5.34·23-s + 3.26·24-s − 2.53·25-s − 4.38·26-s − 15.1·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.702·5-s + 1.33·6-s + 0.377·7-s − 0.353·8-s + 2.54·9-s + 0.496·10-s + 1.01·11-s − 0.941·12-s + 1.21·13-s − 0.267·14-s + 1.32·15-s + 0.250·16-s + 1.69·17-s − 1.79·18-s + 0.285·19-s − 0.351·20-s − 0.711·21-s − 0.717·22-s − 1.11·23-s + 0.665·24-s − 0.506·25-s − 0.859·26-s − 2.90·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 0.714T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 + 9.05T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 9.51T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 7.58T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.72655254136717309631890684968, −6.96346500563172248562163784850, −6.25119303900268399228926065329, −5.76570327541691548479009125700, −5.06055058353714653561740408740, −3.97050762310903014695257653224, −3.59674108812119496385441515916, −1.62853359405487241727000014944, −1.12531580798752930573782687833, 0,
1.12531580798752930573782687833, 1.62853359405487241727000014944, 3.59674108812119496385441515916, 3.97050762310903014695257653224, 5.06055058353714653561740408740, 5.76570327541691548479009125700, 6.25119303900268399228926065329, 6.96346500563172248562163784850, 7.72655254136717309631890684968
