| L(s) = 1 | − 2.30·2-s + 0.482·3-s + 3.30·4-s − 2.32·5-s − 1.10·6-s − 2.51·7-s − 2.99·8-s − 2.76·9-s + 5.35·10-s − 11-s + 1.59·12-s − 1.06·13-s + 5.78·14-s − 1.12·15-s + 0.298·16-s + 1.15·17-s + 6.37·18-s − 4.83·19-s − 7.68·20-s − 1.21·21-s + 2.30·22-s − 4.73·23-s − 1.44·24-s + 0.413·25-s + 2.44·26-s − 2.78·27-s − 8.29·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.278·3-s + 1.65·4-s − 1.04·5-s − 0.453·6-s − 0.949·7-s − 1.05·8-s − 0.922·9-s + 1.69·10-s − 0.301·11-s + 0.459·12-s − 0.294·13-s + 1.54·14-s − 0.289·15-s + 0.0746·16-s + 0.279·17-s + 1.50·18-s − 1.10·19-s − 1.71·20-s − 0.264·21-s + 0.490·22-s − 0.986·23-s − 0.294·24-s + 0.0827·25-s + 0.479·26-s − 0.535·27-s − 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1151270761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1151270761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 233 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 0.482T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 0.714T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.99T + 47T^{2} \) |
| 53 | \( 1 + 1.78T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9.01T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−8.649799015055450325419805905101, −8.267705692685873216790490050919, −7.79334760771134639811221021138, −6.76618042712951918453144663079, −6.31255346016338978976044867836, −4.99007934185186076614976754903, −3.75456422409768099705475745160, −2.96403421872896609308487295751, −1.94661568660454366576750169995, −0.25173095189204418674509245451,
0.25173095189204418674509245451, 1.94661568660454366576750169995, 2.96403421872896609308487295751, 3.75456422409768099705475745160, 4.99007934185186076614976754903, 6.31255346016338978976044867836, 6.76618042712951918453144663079, 7.79334760771134639811221021138, 8.267705692685873216790490050919, 8.649799015055450325419805905101
