| L(s) = 1 | + 2-s + 2.41·3-s + 4-s + 0.585·5-s + 2.41·6-s + 8-s + 2.82·9-s + 0.585·10-s + 11-s + 2.41·12-s + 3.82·13-s + 1.41·15-s + 16-s − 3.65·17-s + 2.82·18-s + 0.585·19-s + 0.585·20-s + 22-s − 6.24·23-s + 2.41·24-s − 4.65·25-s + 3.82·26-s − 0.414·27-s + 2.65·29-s + 1.41·30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.261·5-s + 0.985·6-s + 0.353·8-s + 0.942·9-s + 0.185·10-s + 0.301·11-s + 0.696·12-s + 1.06·13-s + 0.365·15-s + 0.250·16-s − 0.886·17-s + 0.666·18-s + 0.134·19-s + 0.130·20-s + 0.213·22-s − 1.30·23-s + 0.492·24-s − 0.931·25-s + 0.750·26-s − 0.0797·27-s + 0.493·29-s + 0.258·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.123381809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.123381809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−9.818711379786306205475740730765, −8.866034767022388039937337483764, −8.353574591729780356308186853499, −7.43859449054564588581994481337, −6.45956538376357479745880691367, −5.65581137040853651202197418128, −4.26497457519676499926491535625, −3.68887207499182716285174986087, −2.62167723068659257429210932383, −1.71468146781209054614105011130,
1.71468146781209054614105011130, 2.62167723068659257429210932383, 3.68887207499182716285174986087, 4.26497457519676499926491535625, 5.65581137040853651202197418128, 6.45956538376357479745880691367, 7.43859449054564588581994481337, 8.353574591729780356308186853499, 8.866034767022388039937337483764, 9.818711379786306205475740730765
