| L(s) = 1 | − 0.381·2-s + 2.23·3-s − 1.85·4-s − 2.23·5-s − 0.854·6-s + 1.47·8-s + 2.00·9-s + 0.854·10-s − 3·11-s − 4.14·12-s − 13-s − 5.00·15-s + 3.14·16-s − 7.47·17-s − 0.763·18-s + 3·19-s + 4.14·20-s + 1.14·22-s − 3.76·23-s + 3.29·24-s + 0.381·26-s − 2.23·27-s − 4.47·29-s + 1.90·30-s + 5·31-s − 4.14·32-s − 6.70·33-s + ⋯ |
| L(s) = 1 | − 0.270·2-s + 1.29·3-s − 0.927·4-s − 0.999·5-s − 0.348·6-s + 0.520·8-s + 0.666·9-s + 0.270·10-s − 0.904·11-s − 1.19·12-s − 0.277·13-s − 1.29·15-s + 0.786·16-s − 1.81·17-s − 0.180·18-s + 0.688·19-s + 0.927·20-s + 0.244·22-s − 0.784·23-s + 0.671·24-s + 0.0749·26-s − 0.430·27-s − 0.830·29-s + 0.348·30-s + 0.898·31-s − 0.732·32-s − 1.16·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−9.857629142563251110834527806246, −9.129976788151655705340675717080, −8.275967704225302042046045152173, −7.973177014790747387573699419633, −6.99636728548442572942115661332, −5.29432129631228304703014190482, −4.25717014941578661961878362485, −3.50247610759214200813467242100, −2.24242961659627172464937550509, 0,
2.24242961659627172464937550509, 3.50247610759214200813467242100, 4.25717014941578661961878362485, 5.29432129631228304703014190482, 6.99636728548442572942115661332, 7.973177014790747387573699419633, 8.275967704225302042046045152173, 9.129976788151655705340675717080, 9.857629142563251110834527806246
