L(s) = 1 | + 1.68·2-s − 3-s + 0.838·4-s + 0.142·5-s − 1.68·6-s − 0.0282·7-s − 1.95·8-s + 9-s + 0.239·10-s + 0.586·11-s − 0.838·12-s − 13-s − 0.0475·14-s − 0.142·15-s − 4.97·16-s + 6.55·17-s + 1.68·18-s + 3.66·19-s + 0.119·20-s + 0.0282·21-s + 0.988·22-s − 4.69·23-s + 1.95·24-s − 4.97·25-s − 1.68·26-s − 27-s − 0.0236·28-s + ⋯ |
L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.419·4-s + 0.0635·5-s − 0.687·6-s − 0.0106·7-s − 0.692·8-s + 0.333·9-s + 0.0756·10-s + 0.176·11-s − 0.241·12-s − 0.277·13-s − 0.0127·14-s − 0.0366·15-s − 1.24·16-s + 1.58·17-s + 0.397·18-s + 0.839·19-s + 0.0266·20-s + 0.00616·21-s + 0.210·22-s − 0.978·23-s + 0.399·24-s − 0.995·25-s − 0.330·26-s − 0.192·27-s − 0.00447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.613950835\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613950835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 5 | \( 1 - 0.142T + 5T^{2} \) |
| 7 | \( 1 + 0.0282T + 7T^{2} \) |
| 11 | \( 1 - 0.586T + 11T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 - 0.00650T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 + 0.640T + 37T^{2} \) |
| 41 | \( 1 + 0.302T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 - 8.93T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 7.28T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.223211451267048168411648881030, −7.62666210521742465969465357918, −6.65522966683615635284546179136, −5.97447508774920514632288074991, −5.43526385225421081811583030484, −4.78566745291081711971807903052, −3.90082897999479801045639501129, −3.29310620749203784334154137119, −2.19833661768419961218594240069, −0.800322514097457673707539341003,
0.800322514097457673707539341003, 2.19833661768419961218594240069, 3.29310620749203784334154137119, 3.90082897999479801045639501129, 4.78566745291081711971807903052, 5.43526385225421081811583030484, 5.97447508774920514632288074991, 6.65522966683615635284546179136, 7.62666210521742465969465357918, 8.223211451267048168411648881030