| L(s) = 1 | − 2.07·2-s − 2.68·3-s + 2.31·4-s − 2.92·5-s + 5.58·6-s + 0.102·7-s − 0.653·8-s + 4.22·9-s + 6.06·10-s − 11-s − 6.22·12-s − 3.37·13-s − 0.212·14-s + 7.85·15-s − 3.27·16-s − 0.441·17-s − 8.77·18-s − 3.10·19-s − 6.76·20-s − 0.275·21-s + 2.07·22-s + 1.33·23-s + 1.75·24-s + 3.53·25-s + 7.00·26-s − 3.28·27-s + 0.236·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.55·3-s + 1.15·4-s − 1.30·5-s + 2.27·6-s + 0.0386·7-s − 0.231·8-s + 1.40·9-s + 1.91·10-s − 0.301·11-s − 1.79·12-s − 0.934·13-s − 0.0568·14-s + 2.02·15-s − 0.817·16-s − 0.107·17-s − 2.06·18-s − 0.711·19-s − 1.51·20-s − 0.0600·21-s + 0.442·22-s + 0.278·23-s + 0.358·24-s + 0.707·25-s + 1.37·26-s − 0.632·27-s + 0.0447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 - 0.102T + 7T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 0.441T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 4.18T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 0.191T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 + 4.21T + 97T^{2} \) |
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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
−7.49681875511287556746787342835, −6.71372470269060695569487158371, −6.45912006436814179585856584909, −5.21672599303202889392783733513, −4.74300547736823443414555421213, −4.05298473561877205524832970471, −2.84238171116575085735927109567, −1.69191385263842237039830266927, −0.61092203294487883665699662791, 0,
0.61092203294487883665699662791, 1.69191385263842237039830266927, 2.84238171116575085735927109567, 4.05298473561877205524832970471, 4.74300547736823443414555421213, 5.21672599303202889392783733513, 6.45912006436814179585856584909, 6.71372470269060695569487158371, 7.49681875511287556746787342835
