| L(s) = 1 | + 1.92·2-s + 1.68·4-s + 2.68·5-s − 3.03·7-s − 0.597·8-s + 5.15·10-s + 11-s − 1.87·13-s − 5.82·14-s − 4.52·16-s − 5.85·17-s + 2.10·19-s + 4.53·20-s + 1.92·22-s + 5.06·23-s + 2.21·25-s − 3.59·26-s − 5.12·28-s + 0.755·29-s − 7.72·31-s − 7.49·32-s − 11.2·34-s − 8.14·35-s − 8.48·37-s + 4.04·38-s − 1.60·40-s − 7.29·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.844·4-s + 1.20·5-s − 1.14·7-s − 0.211·8-s + 1.63·10-s + 0.301·11-s − 0.519·13-s − 1.55·14-s − 1.13·16-s − 1.42·17-s + 0.482·19-s + 1.01·20-s + 0.409·22-s + 1.05·23-s + 0.442·25-s − 0.705·26-s − 0.968·28-s + 0.140·29-s − 1.38·31-s − 1.32·32-s − 1.92·34-s − 1.37·35-s − 1.39·37-s + 0.655·38-s − 0.253·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 + 7.72T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 + 9.06T + 71T^{2} \) |
| 73 | \( 1 - 0.597T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 + 0.962T + 97T^{2} \) |
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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
−7.15491183841689000616185862879, −6.78337563899629111334334912028, −6.20842882329174414534326998898, −5.41949878720063383713712017553, −5.02825918969520055720701667416, −4.02737691576440546309960437880, −3.29183121759431183923311092174, −2.58690745809306778590631166337, −1.76668059155916390423093867242, 0,
1.76668059155916390423093867242, 2.58690745809306778590631166337, 3.29183121759431183923311092174, 4.02737691576440546309960437880, 5.02825918969520055720701667416, 5.41949878720063383713712017553, 6.20842882329174414534326998898, 6.78337563899629111334334912028, 7.15491183841689000616185862879
