| L(s) = 1 | − 0.221·2-s − 7.95·4-s − 15.1·5-s + 7·7-s + 3.53·8-s + 3.36·10-s − 11·11-s + 37.8·13-s − 1.55·14-s + 62.8·16-s + 61.7·17-s + 54.5·19-s + 120.·20-s + 2.44·22-s − 24.4·23-s + 104.·25-s − 8.38·26-s − 55.6·28-s + 16.2·29-s − 190.·31-s − 42.2·32-s − 13.7·34-s − 106.·35-s + 170.·37-s − 12.1·38-s − 53.6·40-s − 78.0·41-s + ⋯ |
| L(s) = 1 | − 0.0784·2-s − 0.993·4-s − 1.35·5-s + 0.377·7-s + 0.156·8-s + 0.106·10-s − 0.301·11-s + 0.806·13-s − 0.0296·14-s + 0.981·16-s + 0.881·17-s + 0.658·19-s + 1.34·20-s + 0.0236·22-s − 0.222·23-s + 0.838·25-s − 0.0632·26-s − 0.375·28-s + 0.104·29-s − 1.10·31-s − 0.233·32-s − 0.0691·34-s − 0.512·35-s + 0.758·37-s − 0.0516·38-s − 0.212·40-s − 0.297·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.221T + 8T^{2} \) |
| 5 | \( 1 + 15.1T + 125T^{2} \) |
| 13 | \( 1 - 37.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 24.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 16.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 78.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 45.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 650.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 257.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 399.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 198.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 226.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 490.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 221.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
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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
−9.541503303718091858844022581433, −8.621373122572717417447165366548, −7.949288001457336765291419684749, −7.38359355245444053310585557251, −5.86207681602135082004995696436, −4.90853663868747665439950088080, −3.98686009375422435268885149842, −3.26317164093103664574894566988, −1.20295292431872029950449250423, 0,
1.20295292431872029950449250423, 3.26317164093103664574894566988, 3.98686009375422435268885149842, 4.90853663868747665439950088080, 5.86207681602135082004995696436, 7.38359355245444053310585557251, 7.949288001457336765291419684749, 8.621373122572717417447165366548, 9.541503303718091858844022581433
