| L(s) = 1 | − 2.13·2-s − 3-s + 2.55·4-s − 3.00·5-s + 2.13·6-s − 7-s − 1.19·8-s + 9-s + 6.41·10-s − 3.25·11-s − 2.55·12-s − 4.63·13-s + 2.13·14-s + 3.00·15-s − 2.57·16-s − 6.34·17-s − 2.13·18-s − 6.87·19-s − 7.68·20-s + 21-s + 6.95·22-s + 1.21·23-s + 1.19·24-s + 4.02·25-s + 9.88·26-s − 27-s − 2.55·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.577·3-s + 1.27·4-s − 1.34·5-s + 0.871·6-s − 0.377·7-s − 0.420·8-s + 0.333·9-s + 2.02·10-s − 0.981·11-s − 0.738·12-s − 1.28·13-s + 0.570·14-s + 0.775·15-s − 0.643·16-s − 1.53·17-s − 0.503·18-s − 1.57·19-s − 1.71·20-s + 0.218·21-s + 1.48·22-s + 0.252·23-s + 0.242·24-s + 0.805·25-s + 1.93·26-s − 0.192·27-s − 0.483·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 + 0.284T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 1.72T + 61T^{2} \) |
| 67 | \( 1 - 0.210T + 67T^{2} \) |
| 71 | \( 1 - 4.71T + 71T^{2} \) |
| 73 | \( 1 + 3.89T + 73T^{2} \) |
| 79 | \( 1 - 7.83T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 0.135T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−7.63702979102906518744109647345, −6.90117629056242274130305989399, −6.65630206384834371185350567305, −5.42186306413290476610366232934, −4.49966874092047063010845648325, −4.13717492837600752889134193699, −2.69859671568953902113629285013, −2.13895460870714615814948875104, −0.58204385090303638009916465767, 0,
0.58204385090303638009916465767, 2.13895460870714615814948875104, 2.69859671568953902113629285013, 4.13717492837600752889134193699, 4.49966874092047063010845648325, 5.42186306413290476610366232934, 6.65630206384834371185350567305, 6.90117629056242274130305989399, 7.63702979102906518744109647345
