L(s) = 1 | + 2-s − 2.64·3-s + 4-s − 3.70·5-s − 2.64·6-s + 7-s + 8-s + 4.01·9-s − 3.70·10-s + 3.20·11-s − 2.64·12-s − 3.49·13-s + 14-s + 9.81·15-s + 16-s + 2.82·17-s + 4.01·18-s + 5.75·19-s − 3.70·20-s − 2.64·21-s + 3.20·22-s − 3.70·23-s − 2.64·24-s + 8.75·25-s − 3.49·26-s − 2.67·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s − 1.65·5-s − 1.08·6-s + 0.377·7-s + 0.353·8-s + 1.33·9-s − 1.17·10-s + 0.964·11-s − 0.764·12-s − 0.969·13-s + 0.267·14-s + 2.53·15-s + 0.250·16-s + 0.685·17-s + 0.945·18-s + 1.31·19-s − 0.829·20-s − 0.577·21-s + 0.682·22-s − 0.772·23-s − 0.540·24-s + 1.75·25-s − 0.685·26-s − 0.515·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.068316574\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.068316574\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 8.14T + 37T^{2} \) |
| 41 | \( 1 + 4.64T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 - 2.05T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 - 4.16T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−7.63374145929709476962785437632, −7.28850096394812702416385363707, −6.68983020624499328568542788819, −5.71375083742609121292174704156, −5.18275896752657932807035921726, −4.54393539341468371350069833969, −3.85181605458466644869074916591, −3.19664444335616098216753699864, −1.62208676849846711591794510511, −0.54432518119984877666707202427,
0.54432518119984877666707202427, 1.62208676849846711591794510511, 3.19664444335616098216753699864, 3.85181605458466644869074916591, 4.54393539341468371350069833969, 5.18275896752657932807035921726, 5.71375083742609121292174704156, 6.68983020624499328568542788819, 7.28850096394812702416385363707, 7.63374145929709476962785437632