| L(s) = 1 | + 2.47·2-s + 0.982·3-s + 4.13·4-s + 1.35·5-s + 2.43·6-s + 7-s + 5.29·8-s − 2.03·9-s + 3.35·10-s − 3.34·11-s + 4.06·12-s + 2.47·14-s + 1.33·15-s + 4.83·16-s + 0.692·17-s − 5.03·18-s + 7.45·19-s + 5.60·20-s + 0.982·21-s − 8.29·22-s + 8.25·23-s + 5.20·24-s − 3.16·25-s − 4.94·27-s + 4.13·28-s + 1.07·29-s + 3.29·30-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.567·3-s + 2.06·4-s + 0.605·5-s + 0.994·6-s + 0.377·7-s + 1.87·8-s − 0.677·9-s + 1.06·10-s − 1.00·11-s + 1.17·12-s + 0.662·14-s + 0.343·15-s + 1.20·16-s + 0.167·17-s − 1.18·18-s + 1.70·19-s + 1.25·20-s + 0.214·21-s − 1.76·22-s + 1.72·23-s + 1.06·24-s − 0.633·25-s − 0.952·27-s + 0.781·28-s + 0.200·29-s + 0.602·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.743608754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.743608754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 0.982T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 17 | \( 1 - 0.692T + 17T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 - 8.25T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 1.27T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−9.805213723386578421436270972246, −8.979580275000899004613513218170, −7.79371524619998788730698872526, −7.21346420732292525238100588424, −5.96965131283415765414992633480, −5.39156923483981573825831511045, −4.79623644221683867877576492991, −3.37699648603347802633964136418, −2.91754960027362403233379284483, −1.82940525475188076338398329106,
1.82940525475188076338398329106, 2.91754960027362403233379284483, 3.37699648603347802633964136418, 4.79623644221683867877576492991, 5.39156923483981573825831511045, 5.96965131283415765414992633480, 7.21346420732292525238100588424, 7.79371524619998788730698872526, 8.979580275000899004613513218170, 9.805213723386578421436270972246
