| L(s) = 1 | − 1.61·2-s + 0.618·4-s − 5-s + 0.236·7-s + 2.23·8-s + 1.61·10-s − 2·11-s − 3.23·13-s − 0.381·14-s − 4.85·16-s − 0.763·17-s − 2.23·19-s − 0.618·20-s + 3.23·22-s − 5.70·23-s − 4·25-s + 5.23·26-s + 0.145·28-s − 2.76·29-s + 31-s + 3.38·32-s + 1.23·34-s − 0.236·35-s − 2·37-s + 3.61·38-s − 2.23·40-s − 7·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 0.447·5-s + 0.0892·7-s + 0.790·8-s + 0.511·10-s − 0.603·11-s − 0.897·13-s − 0.102·14-s − 1.21·16-s − 0.185·17-s − 0.512·19-s − 0.138·20-s + 0.689·22-s − 1.19·23-s − 0.800·25-s + 1.02·26-s + 0.0275·28-s − 0.513·29-s + 0.179·31-s + 0.597·32-s + 0.211·34-s − 0.0399·35-s − 0.328·37-s + 0.586·38-s − 0.353·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 9.18T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17735342680590771450534303138, −10.20466457484824066484450942809, −9.586556909198595443735209946617, −8.381282397557487660744611065406, −7.83966473501472288475114256122, −6.82468435340708985414018537982, −5.23729792544784815412440799698, −4.02622928327584643718884321074, −2.12507675750278168041227486220, 0,
2.12507675750278168041227486220, 4.02622928327584643718884321074, 5.23729792544784815412440799698, 6.82468435340708985414018537982, 7.83966473501472288475114256122, 8.381282397557487660744611065406, 9.586556909198595443735209946617, 10.20466457484824066484450942809, 11.17735342680590771450534303138
