| L(s) = 1 | − 1.69·2-s − 2.56·3-s + 0.868·4-s + 5-s + 4.33·6-s − 0.819·7-s + 1.91·8-s + 3.56·9-s − 1.69·10-s + 5.38·11-s − 2.22·12-s + 2.46·13-s + 1.38·14-s − 2.56·15-s − 4.98·16-s + 4.20·17-s − 6.03·18-s − 5.38·19-s + 0.868·20-s + 2.09·21-s − 9.12·22-s − 23-s − 4.91·24-s + 25-s − 4.17·26-s − 1.43·27-s − 0.710·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 1.47·3-s + 0.434·4-s + 0.447·5-s + 1.77·6-s − 0.309·7-s + 0.677·8-s + 1.18·9-s − 0.535·10-s + 1.62·11-s − 0.641·12-s + 0.683·13-s + 0.370·14-s − 0.661·15-s − 1.24·16-s + 1.02·17-s − 1.42·18-s − 1.23·19-s + 0.194·20-s + 0.457·21-s − 1.94·22-s − 0.208·23-s − 1.00·24-s + 0.200·25-s − 0.818·26-s − 0.276·27-s − 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4050847023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4050847023\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 + 0.819T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 4.10T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 9.02T + 97T^{2} \) |
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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
−13.39044580534011118362598037612, −12.16639880919803416218905565180, −11.28230148328621455379942078355, −10.33482649441925781624541748506, −9.523336456815885102911488976996, −8.367300205792908111679029711403, −6.74179140659092668547956801214, −6.02785255257524048626341650016, −4.37006479462296182814920270020, −1.14760301018491783932398264453,
1.14760301018491783932398264453, 4.37006479462296182814920270020, 6.02785255257524048626341650016, 6.74179140659092668547956801214, 8.367300205792908111679029711403, 9.523336456815885102911488976996, 10.33482649441925781624541748506, 11.28230148328621455379942078355, 12.16639880919803416218905565180, 13.39044580534011118362598037612
