| L(s) = 1 | + 1.93·2-s − 2.14·3-s + 1.74·4-s − 5-s − 4.14·6-s − 0.502·8-s + 1.60·9-s − 1.93·10-s + 11-s − 3.73·12-s − 5.81·13-s + 2.14·15-s − 4.45·16-s − 3.13·17-s + 3.09·18-s + 4.89·19-s − 1.74·20-s + 1.93·22-s + 9.05·23-s + 1.07·24-s + 25-s − 11.2·26-s + 2.99·27-s + 2.84·29-s + 4.14·30-s − 0.537·31-s − 7.60·32-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 1.23·3-s + 0.870·4-s − 0.447·5-s − 1.69·6-s − 0.177·8-s + 0.533·9-s − 0.611·10-s + 0.301·11-s − 1.07·12-s − 1.61·13-s + 0.553·15-s − 1.11·16-s − 0.759·17-s + 0.730·18-s + 1.12·19-s − 0.389·20-s + 0.412·22-s + 1.88·23-s + 0.220·24-s + 0.200·25-s − 2.20·26-s + 0.577·27-s + 0.527·29-s + 0.757·30-s − 0.0964·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666149742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666149742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 + 0.537T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.74T + 53T^{2} \) |
| 59 | \( 1 + 6.57T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 0.507T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 0.863T + 97T^{2} \) |
| 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
−9.035186795727945474439263386002, −7.68862638700274996491959964784, −6.93970313525444535530777330194, −6.42067377775132958795709562233, −5.41988661240879455886951086309, −4.96754748298805572901957953079, −4.45472178883200490925912283628, −3.32850657396060083015335869252, −2.49181305246848120682799773403, −0.67811422434189062435805349036,
0.67811422434189062435805349036, 2.49181305246848120682799773403, 3.32850657396060083015335869252, 4.45472178883200490925912283628, 4.96754748298805572901957953079, 5.41988661240879455886951086309, 6.42067377775132958795709562233, 6.93970313525444535530777330194, 7.68862638700274996491959964784, 9.035186795727945474439263386002
