| L(s) = 1 | − 0.505·2-s − 2.37·3-s − 7.74·4-s − 5·5-s + 1.20·6-s + 18.6·7-s + 7.96·8-s − 21.3·9-s + 2.52·10-s − 5.43·11-s + 18.3·12-s + 82.2·13-s − 9.43·14-s + 11.8·15-s + 57.9·16-s − 5.47·17-s + 10.7·18-s − 84.2·19-s + 38.7·20-s − 44.3·21-s + 2.74·22-s + 99.9·23-s − 18.9·24-s + 25·25-s − 41.5·26-s + 114.·27-s − 144.·28-s + ⋯ |
| L(s) = 1 | − 0.178·2-s − 0.457·3-s − 0.968·4-s − 0.447·5-s + 0.0817·6-s + 1.00·7-s + 0.351·8-s − 0.791·9-s + 0.0799·10-s − 0.148·11-s + 0.442·12-s + 1.75·13-s − 0.180·14-s + 0.204·15-s + 0.905·16-s − 0.0781·17-s + 0.141·18-s − 1.01·19-s + 0.432·20-s − 0.460·21-s + 0.0266·22-s + 0.905·23-s − 0.160·24-s + 0.200·25-s − 0.313·26-s + 0.818·27-s − 0.974·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 83 | \( 1 - 83T \) |
| good | 2 | \( 1 + 0.505T + 8T^{2} \) |
| 3 | \( 1 + 2.37T + 27T^{2} \) |
| 7 | \( 1 - 18.6T + 343T^{2} \) |
| 11 | \( 1 + 5.43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.47T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 163.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 426.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 24.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 607.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 397.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 381.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 808.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 187.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 623.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 89 | \( 1 + 860.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 844.T + 9.12e5T^{2} \) |
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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
−10.68636884518209164918036301641, −9.181711894994925767496697104898, −8.463935748460651284151531019067, −7.941623396233454124893560751688, −6.41322242428708053234670468313, −5.38191141952732235181332689331, −4.51473488301291803256456707470, −3.41935673563911859619754079960, −1.39209452541155124966103014023, 0,
1.39209452541155124966103014023, 3.41935673563911859619754079960, 4.51473488301291803256456707470, 5.38191141952732235181332689331, 6.41322242428708053234670468313, 7.941623396233454124893560751688, 8.463935748460651284151531019067, 9.181711894994925767496697104898, 10.68636884518209164918036301641
