| L(s) = 1 | − 0.295·2-s + 1.42·3-s − 1.91·4-s + 5-s − 0.420·6-s − 7-s + 1.15·8-s − 0.966·9-s − 0.295·10-s − 0.305·11-s − 2.72·12-s + 6.60·13-s + 0.295·14-s + 1.42·15-s + 3.48·16-s − 3.20·17-s + 0.285·18-s − 3.04·19-s − 1.91·20-s − 1.42·21-s + 0.0902·22-s − 0.304·23-s + 1.64·24-s + 25-s − 1.94·26-s − 5.65·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.208·2-s + 0.823·3-s − 0.956·4-s + 0.447·5-s − 0.171·6-s − 0.377·7-s + 0.408·8-s − 0.322·9-s − 0.0932·10-s − 0.0922·11-s − 0.787·12-s + 1.83·13-s + 0.0788·14-s + 0.368·15-s + 0.871·16-s − 0.776·17-s + 0.0672·18-s − 0.698·19-s − 0.427·20-s − 0.311·21-s + 0.0192·22-s − 0.0634·23-s + 0.336·24-s + 0.200·25-s − 0.382·26-s − 1.08·27-s + 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.295T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 11 | \( 1 + 0.305T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 0.304T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 - 0.438T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 + 4.96T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 8.67T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.67T + 83T^{2} \) |
| 89 | \( 1 - 0.259T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.86230571060068239751652224932, −6.66237084382876422952459771994, −6.19918933990340454899659761209, −5.39449669187828688355691425872, −4.57069089557289762330216798231, −3.71625756467264814676660582711, −3.27724814845066814981743365496, −2.21771875791732132079338372300, −1.29361505796686874344769031300, 0,
1.29361505796686874344769031300, 2.21771875791732132079338372300, 3.27724814845066814981743365496, 3.71625756467264814676660582711, 4.57069089557289762330216798231, 5.39449669187828688355691425872, 6.19918933990340454899659761209, 6.66237084382876422952459771994, 7.86230571060068239751652224932
