L(s) = 1 | − 2.70·2-s + 2.50·3-s + 5.29·4-s − 6.77·6-s − 0.354·7-s − 8.89·8-s + 3.29·9-s + 4.18·11-s + 13.2·12-s + 3.72·13-s + 0.958·14-s + 13.4·16-s + 6.46·17-s − 8.88·18-s − 1.31·19-s − 0.890·21-s − 11.3·22-s + 4.10·23-s − 22.3·24-s − 10.0·26-s + 0.728·27-s − 1.87·28-s − 8.85·29-s + 5.11·31-s − 18.4·32-s + 10.5·33-s − 17.4·34-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 1.44·3-s + 2.64·4-s − 2.76·6-s − 0.134·7-s − 3.14·8-s + 1.09·9-s + 1.26·11-s + 3.83·12-s + 1.03·13-s + 0.256·14-s + 3.35·16-s + 1.56·17-s − 2.09·18-s − 0.300·19-s − 0.194·21-s − 2.41·22-s + 0.856·23-s − 4.55·24-s − 1.97·26-s + 0.140·27-s − 0.355·28-s − 1.64·29-s + 0.918·31-s − 3.26·32-s + 1.82·33-s − 2.99·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.820061803\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820061803\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 0.354T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.673T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 - 9.91T + 97T^{2} \) |
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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
−8.179763805116092928151195459172, −7.76458729862475505946014067873, −7.02077981730225841302149636073, −6.40566406270077212170620869456, −5.56310808258554733405527658971, −3.83074232210901338267723503995, −3.40044102133817410058150930629, −2.51064488947241900811206141122, −1.60136278667572095260678791810, −0.963880784943121677122008610201,
0.963880784943121677122008610201, 1.60136278667572095260678791810, 2.51064488947241900811206141122, 3.40044102133817410058150930629, 3.83074232210901338267723503995, 5.56310808258554733405527658971, 6.40566406270077212170620869456, 7.02077981730225841302149636073, 7.76458729862475505946014067873, 8.179763805116092928151195459172