| L(s) = 1 | − 3.85·2-s + 6.86·4-s − 18.0·5-s − 7·7-s + 4.37·8-s + 69.5·10-s − 11·11-s + 16.8·13-s + 26.9·14-s − 71.7·16-s − 76.0·17-s + 11.8·19-s − 123.·20-s + 42.4·22-s + 175.·23-s + 200.·25-s − 64.9·26-s − 48.0·28-s + 219.·29-s − 214.·31-s + 241.·32-s + 293.·34-s + 126.·35-s + 272.·37-s − 45.8·38-s − 78.9·40-s − 204.·41-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.858·4-s − 1.61·5-s − 0.377·7-s + 0.193·8-s + 2.19·10-s − 0.301·11-s + 0.359·13-s + 0.515·14-s − 1.12·16-s − 1.08·17-s + 0.143·19-s − 1.38·20-s + 0.410·22-s + 1.59·23-s + 1.60·25-s − 0.489·26-s − 0.324·28-s + 1.40·29-s − 1.24·31-s + 1.33·32-s + 1.47·34-s + 0.609·35-s + 1.21·37-s − 0.195·38-s − 0.312·40-s − 0.780·41-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 | \( 1 + 3.85T + 8T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 13 | \( 1 - 16.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 214.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 178.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 238.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 584.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 910.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 387.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 390.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 250.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 54.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 906.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 618.T + 9.12e5T^{2} \) |
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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
−9.368298233270626770570725351583, −8.766854187541899880315934509307, −8.027205823984859827455744637411, −7.29640448502911741119679744593, −6.59266412295047314007437206080, −4.88603513088611670629321519876, −3.95552492277847087508546426587, −2.71817517107884976265953999878, −0.978046534140947832490399795836, 0,
0.978046534140947832490399795836, 2.71817517107884976265953999878, 3.95552492277847087508546426587, 4.88603513088611670629321519876, 6.59266412295047314007437206080, 7.29640448502911741119679744593, 8.027205823984859827455744637411, 8.766854187541899880315934509307, 9.368298233270626770570725351583
