| L(s) = 1 | − 2-s + 0.229·3-s + 4-s + 5-s − 0.229·6-s + 2.13·7-s − 8-s − 2.94·9-s − 10-s + 1.77·11-s + 0.229·12-s − 13-s − 2.13·14-s + 0.229·15-s + 16-s − 1.53·17-s + 2.94·18-s + 4.78·19-s + 20-s + 0.490·21-s − 1.77·22-s + 8.21·23-s − 0.229·24-s + 25-s + 26-s − 1.36·27-s + 2.13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.132·3-s + 0.5·4-s + 0.447·5-s − 0.0936·6-s + 0.807·7-s − 0.353·8-s − 0.982·9-s − 0.316·10-s + 0.535·11-s + 0.0662·12-s − 0.277·13-s − 0.570·14-s + 0.0592·15-s + 0.250·16-s − 0.373·17-s + 0.694·18-s + 1.09·19-s + 0.223·20-s + 0.106·21-s − 0.378·22-s + 1.71·23-s − 0.0468·24-s + 0.200·25-s + 0.196·26-s − 0.262·27-s + 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695160478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695160478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 0.229T + 3T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 37 | \( 1 + 1.15T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 1.55T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 0.795T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 + 4.74T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 7.59T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.646486067828059802317343379658, −7.84387254004365962826656519671, −7.02245555463903552901486700930, −6.45149121591293874139362951048, −5.31183392719905991841292874039, −5.03164284240204088904853310188, −3.60726912258037096980160145525, −2.78102380731806339753881722152, −1.85844225235591564901680990998, −0.846996275616194137084628863858,
0.846996275616194137084628863858, 1.85844225235591564901680990998, 2.78102380731806339753881722152, 3.60726912258037096980160145525, 5.03164284240204088904853310188, 5.31183392719905991841292874039, 6.45149121591293874139362951048, 7.02245555463903552901486700930, 7.84387254004365962826656519671, 8.646486067828059802317343379658
