L(s) = 1 | − 0.898·2-s − 1.87·3-s − 1.19·4-s + 5-s + 1.68·6-s − 2.86·7-s + 2.86·8-s + 0.530·9-s − 0.898·10-s − 0.808·11-s + 2.23·12-s + 5.52·13-s + 2.57·14-s − 1.87·15-s − 0.195·16-s − 6.64·17-s − 0.476·18-s − 2.77·19-s − 1.19·20-s + 5.38·21-s + 0.726·22-s − 5.39·24-s + 25-s − 4.96·26-s + 4.64·27-s + 3.41·28-s − 5.31·29-s + ⋯ |
L(s) = 1 | − 0.635·2-s − 1.08·3-s − 0.595·4-s + 0.447·5-s + 0.689·6-s − 1.08·7-s + 1.01·8-s + 0.176·9-s − 0.284·10-s − 0.243·11-s + 0.646·12-s + 1.53·13-s + 0.688·14-s − 0.485·15-s − 0.0488·16-s − 1.61·17-s − 0.112·18-s − 0.637·19-s − 0.266·20-s + 1.17·21-s + 0.154·22-s − 1.10·24-s + 0.200·25-s − 0.974·26-s + 0.893·27-s + 0.645·28-s − 0.986·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3498217108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3498217108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 0.898T + 2T^{2} \) |
| 3 | \( 1 + 1.87T + 3T^{2} \) |
| 7 | \( 1 + 2.86T + 7T^{2} \) |
| 11 | \( 1 + 0.808T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 + 6.50T + 41T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 + 1.53T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 + 2.09T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 7.47T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 3.68T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−8.649194642804570830312809680704, −8.615217349350265656429672247815, −7.19781025667922052066935276145, −6.33256111969565208386743472311, −6.05568473232084886950074437841, −5.00923948670281989075949020780, −4.23819431461131280212892424382, −3.19224294847468957284663833683, −1.73967056947147036686500252866, −0.43176806090192802427847282011,
0.43176806090192802427847282011, 1.73967056947147036686500252866, 3.19224294847468957284663833683, 4.23819431461131280212892424382, 5.00923948670281989075949020780, 6.05568473232084886950074437841, 6.33256111969565208386743472311, 7.19781025667922052066935276145, 8.615217349350265656429672247815, 8.649194642804570830312809680704