| L(s) = 1 | − 0.554·2-s + 0.801·3-s − 1.69·4-s − 2.80·5-s − 0.445·6-s − 2.69·7-s + 2.04·8-s − 2.35·9-s + 1.55·10-s − 1.19·11-s − 1.35·12-s + 1.49·14-s − 2.24·15-s + 2.24·16-s + 1.13·17-s + 1.30·18-s + 1.93·19-s + 4.74·20-s − 2.15·21-s + 0.664·22-s − 4.60·23-s + 1.64·24-s + 2.85·25-s − 4.29·27-s + 4.55·28-s − 7.89·29-s + 1.24·30-s + ⋯ |
| L(s) = 1 | − 0.392·2-s + 0.462·3-s − 0.846·4-s − 1.25·5-s − 0.181·6-s − 1.01·7-s + 0.724·8-s − 0.785·9-s + 0.491·10-s − 0.361·11-s − 0.391·12-s + 0.399·14-s − 0.580·15-s + 0.561·16-s + 0.275·17-s + 0.308·18-s + 0.444·19-s + 1.06·20-s − 0.471·21-s + 0.141·22-s − 0.959·23-s + 0.335·24-s + 0.570·25-s − 0.826·27-s + 0.860·28-s − 1.46·29-s + 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 3 | \( 1 - 0.801T + 3T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 0.951T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 + 0.0120T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 + 9.25T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 0.807T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 3.13T + 97T^{2} \) |
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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
−12.33536446210603266440589089391, −11.30646866607090638627281997410, −10.02917266805030379807198096797, −9.161774202653594864080465650357, −8.176587484609426278270266087853, −7.50270959552323509370139984132, −5.77643703948707832010929917193, −4.18529951719078164919223936719, −3.17175749010472632307223793978, 0,
3.17175749010472632307223793978, 4.18529951719078164919223936719, 5.77643703948707832010929917193, 7.50270959552323509370139984132, 8.176587484609426278270266087853, 9.161774202653594864080465650357, 10.02917266805030379807198096797, 11.30646866607090638627281997410, 12.33536446210603266440589089391
