| L(s) = 1 | − 2.16·2-s + 2.68·4-s + 3.05·5-s + 3.25·7-s − 1.48·8-s − 6.62·10-s − 11-s − 1.80·13-s − 7.05·14-s − 2.15·16-s + 0.205·17-s − 0.465·19-s + 8.21·20-s + 2.16·22-s − 6.67·23-s + 4.35·25-s + 3.90·26-s + 8.75·28-s − 9.18·29-s − 1.01·31-s + 7.63·32-s − 0.445·34-s + 9.95·35-s + 2.09·37-s + 1.00·38-s − 4.55·40-s − 0.539·41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.34·4-s + 1.36·5-s + 1.23·7-s − 0.526·8-s − 2.09·10-s − 0.301·11-s − 0.500·13-s − 1.88·14-s − 0.537·16-s + 0.0498·17-s − 0.106·19-s + 1.83·20-s + 0.461·22-s − 1.39·23-s + 0.870·25-s + 0.766·26-s + 1.65·28-s − 1.70·29-s − 0.182·31-s + 1.34·32-s − 0.0763·34-s + 1.68·35-s + 0.344·37-s + 0.163·38-s − 0.720·40-s − 0.0842·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 0.205T + 17T^{2} \) |
| 19 | \( 1 + 0.465T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 + 9.18T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 + 0.539T + 41T^{2} \) |
| 43 | \( 1 + 9.03T + 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 0.483T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 0.323T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−7.82420589734870791855297668581, −6.98984989404773568782419848765, −6.28800140496219717806004499924, −5.43350107686310902395021671363, −4.95318726166303256048359926949, −3.86932206069721637288735127768, −2.43810768548017332467144379573, −1.94118069482642606239349094110, −1.38712300863487446069668996945, 0,
1.38712300863487446069668996945, 1.94118069482642606239349094110, 2.43810768548017332467144379573, 3.86932206069721637288735127768, 4.95318726166303256048359926949, 5.43350107686310902395021671363, 6.28800140496219717806004499924, 6.98984989404773568782419848765, 7.82420589734870791855297668581
