| L(s) = 1 | + 1.12·2-s − 0.730·4-s + 0.792·5-s + 3.80·7-s − 3.07·8-s + 0.893·10-s − 11-s − 13-s + 4.28·14-s − 2.00·16-s + 3.83·17-s + 7.92·19-s − 0.578·20-s − 1.12·22-s + 2.44·23-s − 4.37·25-s − 1.12·26-s − 2.78·28-s + 2.56·29-s − 2.32·31-s + 3.89·32-s + 4.32·34-s + 3.01·35-s − 6.09·37-s + 8.92·38-s − 2.43·40-s + 7.74·41-s + ⋯ |
| L(s) = 1 | + 0.796·2-s − 0.365·4-s + 0.354·5-s + 1.43·7-s − 1.08·8-s + 0.282·10-s − 0.301·11-s − 0.277·13-s + 1.14·14-s − 0.501·16-s + 0.931·17-s + 1.81·19-s − 0.129·20-s − 0.240·22-s + 0.510·23-s − 0.874·25-s − 0.220·26-s − 0.525·28-s + 0.476·29-s − 0.417·31-s + 0.688·32-s + 0.741·34-s + 0.510·35-s − 1.00·37-s + 1.44·38-s − 0.385·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.628376350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628376350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 5 | \( 1 - 0.792T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 + 4.49T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 3.48T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−9.584057727537478378821576406670, −8.932364830014988643573624934830, −7.88558038219852760392008172634, −7.38039859871136099928059262951, −5.87785210342673564396058842013, −5.34806630434486280765144302286, −4.71546980214128540903962171124, −3.68587779026967431876244526780, −2.60592575878717677758071205735, −1.17583566109110072908496266946,
1.17583566109110072908496266946, 2.60592575878717677758071205735, 3.68587779026967431876244526780, 4.71546980214128540903962171124, 5.34806630434486280765144302286, 5.87785210342673564396058842013, 7.38039859871136099928059262951, 7.88558038219852760392008172634, 8.932364830014988643573624934830, 9.584057727537478378821576406670
