| L(s) = 1 | + 0.435·3-s + 5-s − 2.87·7-s − 2.81·9-s − 2.29·11-s − 0.0953·13-s + 0.435·15-s + 3.92·17-s + 6.59·19-s − 1.25·21-s − 0.150·23-s + 25-s − 2.53·27-s + 2.49·29-s − 0.535·31-s − 1.00·33-s − 2.87·35-s − 11.7·37-s − 0.0415·39-s + 9.05·41-s + 8.32·43-s − 2.81·45-s − 1.88·47-s + 1.28·49-s + 1.70·51-s + 10.4·53-s − 2.29·55-s + ⋯ |
| L(s) = 1 | + 0.251·3-s + 0.447·5-s − 1.08·7-s − 0.936·9-s − 0.692·11-s − 0.0264·13-s + 0.112·15-s + 0.951·17-s + 1.51·19-s − 0.273·21-s − 0.0313·23-s + 0.200·25-s − 0.487·27-s + 0.464·29-s − 0.0961·31-s − 0.174·33-s − 0.486·35-s − 1.93·37-s − 0.00664·39-s + 1.41·41-s + 1.26·43-s − 0.418·45-s − 0.274·47-s + 0.184·49-s + 0.239·51-s + 1.42·53-s − 0.309·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 0.435T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 0.0953T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 23 | \( 1 + 0.150T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 + 0.535T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 2.46T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 5.10T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.59607655436493967387305959244, −7.18033561033395158671778157360, −6.08252642699696339666069114007, −5.69728467221211843213540883357, −5.04448065750752537969112394916, −3.82619987244305728036349684587, −2.98678416882571954384076988320, −2.69655819426724516971941966233, −1.29168242660714256757777483219, 0,
1.29168242660714256757777483219, 2.69655819426724516971941966233, 2.98678416882571954384076988320, 3.82619987244305728036349684587, 5.04448065750752537969112394916, 5.69728467221211843213540883357, 6.08252642699696339666069114007, 7.18033561033395158671778157360, 7.59607655436493967387305959244
