| L(s) = 1 | − 1.52·2-s − 5.66·4-s − 19.3·5-s + 4.84·7-s + 20.8·8-s + 29.5·10-s + 61.0·11-s + 13·13-s − 7.39·14-s + 13.5·16-s + 41.7·17-s − 107.·19-s + 109.·20-s − 93.2·22-s − 28.5·23-s + 249.·25-s − 19.8·26-s − 27.4·28-s + 89.8·29-s + 183.·31-s − 187.·32-s − 63.7·34-s − 93.6·35-s + 418.·37-s + 164.·38-s − 403.·40-s + 142.·41-s + ⋯ |
| L(s) = 1 | − 0.539·2-s − 0.708·4-s − 1.72·5-s + 0.261·7-s + 0.922·8-s + 0.933·10-s + 1.67·11-s + 0.277·13-s − 0.141·14-s + 0.211·16-s + 0.596·17-s − 1.29·19-s + 1.22·20-s − 0.903·22-s − 0.258·23-s + 1.99·25-s − 0.149·26-s − 0.185·28-s + 0.575·29-s + 1.06·31-s − 1.03·32-s − 0.321·34-s − 0.452·35-s + 1.85·37-s + 0.700·38-s − 1.59·40-s + 0.543·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7685428724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7685428724\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 - 4.84T + 343T^{2} \) |
| 11 | \( 1 - 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−12.87296648920289278079145791866, −11.86189749063646866025045070306, −11.09264079525447608636223743451, −9.725379415936590215919511415943, −8.517160621733974747546508666483, −7.996118490510139133463219761213, −6.62471132890081211081095078756, −4.51469871805688857575290619436, −3.80091103052839321503528647309, −0.858882361545029268414711747487,
0.858882361545029268414711747487, 3.80091103052839321503528647309, 4.51469871805688857575290619436, 6.62471132890081211081095078756, 7.996118490510139133463219761213, 8.517160621733974747546508666483, 9.725379415936590215919511415943, 11.09264079525447608636223743451, 11.86189749063646866025045070306, 12.87296648920289278079145791866
