| L(s) = 1 | − 2-s + 0.237·3-s + 4-s + 0.796·5-s − 0.237·6-s + 2.80·7-s − 8-s − 2.94·9-s − 0.796·10-s + 4.74·11-s + 0.237·12-s − 2.18·13-s − 2.80·14-s + 0.188·15-s + 16-s + 6.02·17-s + 2.94·18-s − 19-s + 0.796·20-s + 0.664·21-s − 4.74·22-s − 6.41·23-s − 0.237·24-s − 4.36·25-s + 2.18·26-s − 1.40·27-s + 2.80·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.136·3-s + 0.5·4-s + 0.356·5-s − 0.0967·6-s + 1.05·7-s − 0.353·8-s − 0.981·9-s − 0.251·10-s + 1.42·11-s + 0.0684·12-s − 0.605·13-s − 0.749·14-s + 0.0487·15-s + 0.250·16-s + 1.46·17-s + 0.693·18-s − 0.229·19-s + 0.178·20-s + 0.145·21-s − 1.01·22-s − 1.33·23-s − 0.0483·24-s − 0.873·25-s + 0.428·26-s − 0.271·27-s + 0.529·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924166078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924166078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.237T + 3T^{2} \) |
| 5 | \( 1 - 0.796T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 23 | \( 1 + 6.41T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.792T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 4.63T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 1.83T + 71T^{2} \) |
| 73 | \( 1 - 0.274T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−7.915510090584704222261234688004, −7.45392645469597861660412683206, −6.29177309778338652742128913574, −6.00237109421526014274212914877, −5.14402988237999195463605847698, −4.24542132356743219683060912394, −3.42106863397722750403244938415, −2.40829127201251788836526079025, −1.71755318967417232458525934728, −0.78458632159762089654567952204,
0.78458632159762089654567952204, 1.71755318967417232458525934728, 2.40829127201251788836526079025, 3.42106863397722750403244938415, 4.24542132356743219683060912394, 5.14402988237999195463605847698, 6.00237109421526014274212914877, 6.29177309778338652742128913574, 7.45392645469597861660412683206, 7.915510090584704222261234688004
