L(s) = 1 | + 1.79·2-s − 3-s + 1.20·4-s − 4.32·5-s − 1.79·6-s − 1.42·8-s + 9-s − 7.74·10-s − 0.0430·11-s − 1.20·12-s − 13-s + 4.32·15-s − 4.95·16-s + 7.25·17-s + 1.79·18-s − 5.95·19-s − 5.21·20-s − 0.0770·22-s + 7.94·23-s + 1.42·24-s + 13.7·25-s − 1.79·26-s − 27-s − 7.11·29-s + 7.74·30-s + 2.28·31-s − 6.03·32-s + ⋯ |
L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.602·4-s − 1.93·5-s − 0.730·6-s − 0.503·8-s + 0.333·9-s − 2.44·10-s − 0.0129·11-s − 0.347·12-s − 0.277·13-s + 1.11·15-s − 1.23·16-s + 1.75·17-s + 0.421·18-s − 1.36·19-s − 1.16·20-s − 0.0164·22-s + 1.65·23-s + 0.290·24-s + 2.74·25-s − 0.351·26-s − 0.192·27-s − 1.32·29-s + 1.41·30-s + 0.409·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.488789196\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.488789196\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 + 4.32T + 5T^{2} \) |
| 11 | \( 1 + 0.0430T + 11T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 9.16T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 0.368T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 - 0.658T + 83T^{2} \) |
| 89 | \( 1 - 5.16T + 89T^{2} \) |
| 97 | \( 1 + 2.35T + 97T^{2} \) |
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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
−9.067065005906860258629870059149, −8.252539623316649012461197025692, −7.40181376809084424392822054334, −6.82760673255155642811886888043, −5.73848339140696579582327232045, −5.01001199934170250875267615732, −4.23133084320399953753082400746, −3.67943120775441208670342152324, −2.79814166847247100068046144426, −0.68884820551918585002336497459,
0.68884820551918585002336497459, 2.79814166847247100068046144426, 3.67943120775441208670342152324, 4.23133084320399953753082400746, 5.01001199934170250875267615732, 5.73848339140696579582327232045, 6.82760673255155642811886888043, 7.40181376809084424392822054334, 8.252539623316649012461197025692, 9.067065005906860258629870059149