L(s) = 1 | − 2-s + 2.29·3-s + 4-s + 0.901·5-s − 2.29·6-s − 7-s − 8-s + 2.24·9-s − 0.901·10-s − 4.33·11-s + 2.29·12-s + 14-s + 2.06·15-s + 16-s + 5.06·17-s − 2.24·18-s + 6.17·19-s + 0.901·20-s − 2.29·21-s + 4.33·22-s + 8.45·23-s − 2.29·24-s − 4.18·25-s − 1.72·27-s − 28-s − 2.19·29-s − 2.06·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.403·5-s − 0.935·6-s − 0.377·7-s − 0.353·8-s + 0.749·9-s − 0.285·10-s − 1.30·11-s + 0.661·12-s + 0.267·14-s + 0.533·15-s + 0.250·16-s + 1.22·17-s − 0.529·18-s + 1.41·19-s + 0.201·20-s − 0.499·21-s + 0.924·22-s + 1.76·23-s − 0.467·24-s − 0.837·25-s − 0.331·27-s − 0.188·28-s − 0.407·29-s − 0.377·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.135202534\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.135202534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 0.901T + 5T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 0.873T + 31T^{2} \) |
| 37 | \( 1 - 0.144T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 1.69T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + 0.539T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.014961012948954880357290749550, −8.245295601219606632413442463455, −7.55521684628330240151612678599, −7.17938095192006514025615122306, −5.78818877419638787665568223639, −5.23826470791764168313747418806, −3.69349810585511983968602684230, −2.93419285884962137374528137879, −2.35344231081615429680090812760, −1.00563473993145931546257214403,
1.00563473993145931546257214403, 2.35344231081615429680090812760, 2.93419285884962137374528137879, 3.69349810585511983968602684230, 5.23826470791764168313747418806, 5.78818877419638787665568223639, 7.17938095192006514025615122306, 7.55521684628330240151612678599, 8.245295601219606632413442463455, 9.014961012948954880357290749550