| L(s) = 1 | − 0.262·2-s + 3-s − 1.93·4-s + 5-s − 0.262·6-s − 3.19·7-s + 1.03·8-s + 9-s − 0.262·10-s − 1.93·12-s − 1.11·13-s + 0.837·14-s + 15-s + 3.59·16-s − 0.0882·17-s − 0.262·18-s − 0.0688·19-s − 1.93·20-s − 3.19·21-s + 6.65·23-s + 1.03·24-s + 25-s + 0.293·26-s + 27-s + 6.16·28-s + 3.73·29-s − 0.262·30-s + ⋯ |
| L(s) = 1 | − 0.185·2-s + 0.577·3-s − 0.965·4-s + 0.447·5-s − 0.107·6-s − 1.20·7-s + 0.364·8-s + 0.333·9-s − 0.0829·10-s − 0.557·12-s − 0.310·13-s + 0.223·14-s + 0.258·15-s + 0.897·16-s − 0.0213·17-s − 0.0618·18-s − 0.0157·19-s − 0.431·20-s − 0.696·21-s + 1.38·23-s + 0.210·24-s + 0.200·25-s + 0.0576·26-s + 0.192·27-s + 1.16·28-s + 0.693·29-s − 0.0479·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.262T + 2T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 + 0.0882T + 17T^{2} \) |
| 19 | \( 1 + 0.0688T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.58T + 31T^{2} \) |
| 37 | \( 1 + 8.33T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 0.908T + 47T^{2} \) |
| 53 | \( 1 - 0.872T + 53T^{2} \) |
| 59 | \( 1 - 1.83T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 - 0.791T + 79T^{2} \) |
| 83 | \( 1 - 0.247T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−8.885814841529072785854325944869, −8.432161430526211979113505024078, −7.22533755633391088153093552844, −6.67929418610635142023527247627, −5.49330806667565481528107665989, −4.80968967207498363703640685356, −3.60135474851519164453162302319, −3.05234755630504328191036398206, −1.60241143058213505606575442354, 0,
1.60241143058213505606575442354, 3.05234755630504328191036398206, 3.60135474851519164453162302319, 4.80968967207498363703640685356, 5.49330806667565481528107665989, 6.67929418610635142023527247627, 7.22533755633391088153093552844, 8.432161430526211979113505024078, 8.885814841529072785854325944869
