| L(s) = 1 | − 1.21·2-s − 3-s − 0.522·4-s − 5-s + 1.21·6-s − 3.08·7-s + 3.06·8-s + 9-s + 1.21·10-s − 4.01·11-s + 0.522·12-s + 1.19·13-s + 3.75·14-s + 15-s − 2.68·16-s + 2.35·17-s − 1.21·18-s − 3.32·19-s + 0.522·20-s + 3.08·21-s + 4.87·22-s − 3.06·24-s + 25-s − 1.44·26-s − 27-s + 1.61·28-s + 5.62·29-s + ⋯ |
| L(s) = 1 | − 0.859·2-s − 0.577·3-s − 0.261·4-s − 0.447·5-s + 0.496·6-s − 1.16·7-s + 1.08·8-s + 0.333·9-s + 0.384·10-s − 1.21·11-s + 0.150·12-s + 0.330·13-s + 1.00·14-s + 0.258·15-s − 0.670·16-s + 0.570·17-s − 0.286·18-s − 0.762·19-s + 0.116·20-s + 0.674·21-s + 1.04·22-s − 0.625·24-s + 0.200·25-s − 0.283·26-s − 0.192·27-s + 0.304·28-s + 1.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2101476984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2101476984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 29 | \( 1 - 5.62T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 - 0.158T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 2.60T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 - 1.86T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.107356845859942315178145825280, −7.14236180126285416900443858050, −6.65603997789744587778114019504, −5.81835426765206233058013016924, −5.03398865431317974529963770954, −4.38137144271132839757990489990, −3.48308336465324034881747780359, −2.67961135604308245352715995503, −1.37685877617096320638039472662, −0.28602879745979933113313531084,
0.28602879745979933113313531084, 1.37685877617096320638039472662, 2.67961135604308245352715995503, 3.48308336465324034881747780359, 4.38137144271132839757990489990, 5.03398865431317974529963770954, 5.81835426765206233058013016924, 6.65603997789744587778114019504, 7.14236180126285416900443858050, 8.107356845859942315178145825280
