| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s − 20.0·7-s − 8·8-s + 9·9-s + 10·10-s − 16.6·11-s − 12·12-s + 56.7·13-s + 40.0·14-s + 15·15-s + 16·16-s + 109.·17-s − 18·18-s + 19·19-s − 20·20-s + 60.1·21-s + 33.3·22-s − 44.6·23-s + 24·24-s + 25·25-s − 113.·26-s − 27·27-s − 80.1·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.08·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.456·11-s − 0.288·12-s + 1.21·13-s + 0.765·14-s + 0.258·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.624·21-s + 0.322·22-s − 0.405·23-s + 0.204·24-s + 0.200·25-s − 0.856·26-s − 0.192·27-s − 0.541·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 19 | \( 1 - 19T \) |
| good | 7 | \( 1 + 20.0T + 343T^{2} \) |
| 11 | \( 1 + 16.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 44.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 162.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 0.878T + 1.03e5T^{2} \) |
| 53 | \( 1 - 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 863.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 131.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 306.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 312.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 661.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 6.12T + 4.93e5T^{2} \) |
| 83 | \( 1 + 992.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−9.871982365083530925021238674552, −9.193323999280936699257954786268, −8.002853753850172375728771397464, −7.34965948261003075090525560503, −6.19552686049794967455417080994, −5.61816174379104197710632048654, −3.96157533766119271749753722639, −3.00453949805353343623814577561, −1.21311560557006009794009683142, 0,
1.21311560557006009794009683142, 3.00453949805353343623814577561, 3.96157533766119271749753722639, 5.61816174379104197710632048654, 6.19552686049794967455417080994, 7.34965948261003075090525560503, 8.002853753850172375728771397464, 9.193323999280936699257954786268, 9.871982365083530925021238674552
