| L(s) = 1 | − 2.67·2-s − 2.54·3-s + 5.14·4-s − 3.71·5-s + 6.80·6-s + 3.32·7-s − 8.42·8-s + 3.48·9-s + 9.93·10-s − 0.879·11-s − 13.1·12-s + 2.32·13-s − 8.87·14-s + 9.46·15-s + 12.2·16-s + 6.68·17-s − 9.32·18-s + 7.23·19-s − 19.1·20-s − 8.45·21-s + 2.35·22-s + 1.22·23-s + 21.4·24-s + 8.80·25-s − 6.21·26-s − 1.23·27-s + 17.0·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.47·3-s + 2.57·4-s − 1.66·5-s + 2.78·6-s + 1.25·7-s − 2.97·8-s + 1.16·9-s + 3.14·10-s − 0.265·11-s − 3.78·12-s + 0.645·13-s − 2.37·14-s + 2.44·15-s + 3.05·16-s + 1.62·17-s − 2.19·18-s + 1.65·19-s − 4.27·20-s − 1.84·21-s + 0.501·22-s + 0.255·23-s + 4.37·24-s + 1.76·25-s − 1.21·26-s − 0.238·27-s + 3.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 0.879T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 + 0.845T + 29T^{2} \) |
| 31 | \( 1 - 0.0194T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 + 8.06T + 83T^{2} \) |
| 89 | \( 1 - 3.48T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.51526829930039523660700865381, −7.27908504386444621743322742184, −6.36401492492957492057370416286, −5.44343748879958731765087712585, −5.00602268816235634468979771886, −3.78826158566653816657955830049, −2.97674143945116614757091811732, −1.36739674123270298761629297529, −1.01565675800521007828934358585, 0,
1.01565675800521007828934358585, 1.36739674123270298761629297529, 2.97674143945116614757091811732, 3.78826158566653816657955830049, 5.00602268816235634468979771886, 5.44343748879958731765087712585, 6.36401492492957492057370416286, 7.27908504386444621743322742184, 7.51526829930039523660700865381
