| L(s) = 1 | + 0.351·2-s − 2.89·3-s − 1.87·4-s + 2.97·5-s − 1.01·6-s − 1.08·7-s − 1.36·8-s + 5.37·9-s + 1.04·10-s − 2.43·11-s + 5.42·12-s + 2.87·13-s − 0.380·14-s − 8.60·15-s + 3.27·16-s − 1.82·17-s + 1.88·18-s + 2.11·19-s − 5.57·20-s + 3.13·21-s − 0.854·22-s − 0.442·23-s + 3.94·24-s + 3.84·25-s + 1.00·26-s − 6.86·27-s + 2.03·28-s + ⋯ |
| L(s) = 1 | + 0.248·2-s − 1.67·3-s − 0.938·4-s + 1.32·5-s − 0.415·6-s − 0.409·7-s − 0.481·8-s + 1.79·9-s + 0.330·10-s − 0.733·11-s + 1.56·12-s + 0.797·13-s − 0.101·14-s − 2.22·15-s + 0.818·16-s − 0.442·17-s + 0.444·18-s + 0.485·19-s − 1.24·20-s + 0.683·21-s − 0.182·22-s − 0.0922·23-s + 0.804·24-s + 0.768·25-s + 0.198·26-s − 1.32·27-s + 0.383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 - 0.351T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 + 0.442T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 8.41T + 43T^{2} \) |
| 47 | \( 1 + 7.96T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 6.43T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 2.25T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−9.785759918134305489468905270899, −9.106566085501888145015611426391, −7.87948169211071985948804159612, −6.43796728442677879712916188352, −6.12017979766790521169203946941, −5.23145597385969311379387948193, −4.76215951198312336723481108109, −3.34003095594142756711301675214, −1.53693472626597640772997237449, 0,
1.53693472626597640772997237449, 3.34003095594142756711301675214, 4.76215951198312336723481108109, 5.23145597385969311379387948193, 6.12017979766790521169203946941, 6.43796728442677879712916188352, 7.87948169211071985948804159612, 9.106566085501888145015611426391, 9.785759918134305489468905270899
