| L(s) = 1 | − 2.18·2-s − 3-s + 2.76·4-s − 5-s + 2.18·6-s + 0.121·7-s − 1.67·8-s + 9-s + 2.18·10-s − 2.69·11-s − 2.76·12-s − 13-s − 0.265·14-s + 15-s − 1.88·16-s + 4.36·17-s − 2.18·18-s + 0.489·19-s − 2.76·20-s − 0.121·21-s + 5.89·22-s − 1.37·23-s + 1.67·24-s + 25-s + 2.18·26-s − 27-s + 0.336·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.577·3-s + 1.38·4-s − 0.447·5-s + 0.891·6-s + 0.0460·7-s − 0.590·8-s + 0.333·9-s + 0.690·10-s − 0.813·11-s − 0.798·12-s − 0.277·13-s − 0.0710·14-s + 0.258·15-s − 0.471·16-s + 1.05·17-s − 0.514·18-s + 0.112·19-s − 0.618·20-s − 0.0265·21-s + 1.25·22-s − 0.287·23-s + 0.340·24-s + 0.200·25-s + 0.428·26-s − 0.192·27-s + 0.0636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 7 | \( 1 - 0.121T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 - 0.489T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 + 0.663T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 0.377T + 67T^{2} \) |
| 71 | \( 1 - 0.664T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 + 0.171T + 83T^{2} \) |
| 89 | \( 1 + 8.62T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.922159316561751941864068765696, −7.17686736135789184412790051030, −6.69175964792816117010791530266, −5.60496218116120063600021064791, −5.03636254769466195982095553943, −4.01815920058029221858764800062, −2.97535514752658520580369929769, −1.96217000567855089876505388832, −0.947107918539108185191879621888, 0,
0.947107918539108185191879621888, 1.96217000567855089876505388832, 2.97535514752658520580369929769, 4.01815920058029221858764800062, 5.03636254769466195982095553943, 5.60496218116120063600021064791, 6.69175964792816117010791530266, 7.17686736135789184412790051030, 7.922159316561751941864068765696
