L(s) = 1 | − 0.785·2-s − 0.0947·3-s − 1.38·4-s + 0.0744·6-s + 3.94·7-s + 2.65·8-s − 2.99·9-s + 4.77·11-s + 0.130·12-s − 2.97·13-s − 3.09·14-s + 0.675·16-s − 1.89·17-s + 2.35·18-s + 1.88·19-s − 0.373·21-s − 3.75·22-s + 6.71·23-s − 0.251·24-s + 2.33·26-s + 0.567·27-s − 5.45·28-s + 1.28·29-s + 8.34·31-s − 5.84·32-s − 0.452·33-s + 1.48·34-s + ⋯ |
L(s) = 1 | − 0.555·2-s − 0.0546·3-s − 0.691·4-s + 0.0303·6-s + 1.49·7-s + 0.939·8-s − 0.997·9-s + 1.43·11-s + 0.0377·12-s − 0.825·13-s − 0.828·14-s + 0.168·16-s − 0.458·17-s + 0.554·18-s + 0.433·19-s − 0.0815·21-s − 0.799·22-s + 1.40·23-s − 0.0513·24-s + 0.458·26-s + 0.109·27-s − 1.03·28-s + 0.238·29-s + 1.49·31-s − 1.03·32-s − 0.0787·33-s + 0.254·34-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
\(L(1)\) |
\(\approx\) |
\(1.470237235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.470237235\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.785T + 2T^{2} \) |
| 3 | \( 1 + 0.0947T + 3T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 5.77T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 3.20T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 1.34T + 97T^{2} \) |
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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
−8.403732320278815651116234866872, −7.43192296379604171528731274055, −6.93121394492595304673738259413, −5.82621392091077491201535581345, −5.02143271416662033566678514347, −4.63530741936935220510165295990, −3.77543789946576099580353359208, −2.64916469504300018559997624355, −1.56080090002503691647516158812, −0.76205925320478909390846390238,
0.76205925320478909390846390238, 1.56080090002503691647516158812, 2.64916469504300018559997624355, 3.77543789946576099580353359208, 4.63530741936935220510165295990, 5.02143271416662033566678514347, 5.82621392091077491201535581345, 6.93121394492595304673738259413, 7.43192296379604171528731274055, 8.403732320278815651116234866872