| L(s) = 1 | − 4.44·2-s + 11.7·4-s + 22.0·5-s + 7·7-s − 16.6·8-s − 97.8·10-s − 11·11-s − 51.5·13-s − 31.1·14-s − 19.9·16-s + 26.5·17-s + 99.6·19-s + 258.·20-s + 48.8·22-s − 28.1·23-s + 359.·25-s + 229.·26-s + 82.2·28-s + 43.9·29-s − 83.8·31-s + 221.·32-s − 118.·34-s + 154.·35-s + 306.·37-s − 442.·38-s − 366.·40-s − 200.·41-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.46·4-s + 1.96·5-s + 0.377·7-s − 0.736·8-s − 3.09·10-s − 0.301·11-s − 1.10·13-s − 0.593·14-s − 0.311·16-s + 0.378·17-s + 1.20·19-s + 2.89·20-s + 0.473·22-s − 0.255·23-s + 2.87·25-s + 1.72·26-s + 0.555·28-s + 0.281·29-s − 0.485·31-s + 1.22·32-s − 0.595·34-s + 0.744·35-s + 1.36·37-s − 1.89·38-s − 1.44·40-s − 0.765·41-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
| \(L(2)\) |
\(\approx\) |
\(1.424556137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424556137\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.44T + 8T^{2} \) |
| 5 | \( 1 - 22.0T + 125T^{2} \) |
| 13 | \( 1 + 51.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 13.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 308.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 622.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 87.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 608.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 464.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 255.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 261.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 953.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 839.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 349.T + 9.12e5T^{2} \) |
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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
−9.741841940155213333084476005904, −9.552887722902233487022643032276, −8.547530456429264241022975614183, −7.58793430162303772178531478675, −6.79230473259288939994217007188, −5.71884222139161686345200904551, −4.92934828404225892461492066174, −2.71850120161744464841518123254, −1.92126475152865460232851460853, −0.907942599976805648096740243873,
0.907942599976805648096740243873, 1.92126475152865460232851460853, 2.71850120161744464841518123254, 4.92934828404225892461492066174, 5.71884222139161686345200904551, 6.79230473259288939994217007188, 7.58793430162303772178531478675, 8.547530456429264241022975614183, 9.552887722902233487022643032276, 9.741841940155213333084476005904
