| L(s) = 1 | + 1.26·2-s + 1.48·3-s − 0.395·4-s − 3.80·5-s + 1.88·6-s + 2.18·7-s − 3.03·8-s − 0.796·9-s − 4.81·10-s − 0.950·11-s − 0.587·12-s + 0.168·13-s + 2.77·14-s − 5.64·15-s − 3.05·16-s − 6.57·17-s − 1.00·18-s − 1.15·19-s + 1.50·20-s + 3.24·21-s − 1.20·22-s − 4.62·23-s − 4.50·24-s + 9.44·25-s + 0.212·26-s − 5.63·27-s − 0.866·28-s + ⋯ |
| L(s) = 1 | + 0.895·2-s + 0.856·3-s − 0.197·4-s − 1.69·5-s + 0.767·6-s + 0.827·7-s − 1.07·8-s − 0.265·9-s − 1.52·10-s − 0.286·11-s − 0.169·12-s + 0.0466·13-s + 0.741·14-s − 1.45·15-s − 0.762·16-s − 1.59·17-s − 0.237·18-s − 0.264·19-s + 0.336·20-s + 0.709·21-s − 0.256·22-s − 0.965·23-s − 0.919·24-s + 1.88·25-s + 0.0417·26-s − 1.08·27-s − 0.163·28-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 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 0.950T + 11T^{2} \) |
| 13 | \( 1 - 0.168T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 - 0.328T + 41T^{2} \) |
| 43 | \( 1 - 9.63T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 + 7.33T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 - 0.552T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 + 7.92T + 73T^{2} \) |
| 79 | \( 1 - 4.54T + 79T^{2} \) |
| 83 | \( 1 - 0.326T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−9.258806756113854798518371431900, −8.543788303487334718790899149991, −8.115241750757262882905176500292, −7.25153795984165652669876994301, −6.01420135431508684921074847085, −4.75635777463472870802826509276, −4.23105578159503209608378092609, −3.44517747570623331972260076133, −2.39710487264412169029959229348, 0,
2.39710487264412169029959229348, 3.44517747570623331972260076133, 4.23105578159503209608378092609, 4.75635777463472870802826509276, 6.01420135431508684921074847085, 7.25153795984165652669876994301, 8.115241750757262882905176500292, 8.543788303487334718790899149991, 9.258806756113854798518371431900
