| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s − 15·7-s + 8·8-s + 9·9-s + 10·10-s − 50.2·11-s + 12·12-s + 21.0·13-s − 30·14-s + 15·15-s + 16·16-s − 52.0·17-s + 18·18-s − 101.·19-s + 20·20-s − 45·21-s − 100.·22-s + 37.5·23-s + 24·24-s + 25·25-s + 42.1·26-s + 27·27-s − 60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.809·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.37·11-s + 0.288·12-s + 0.449·13-s − 0.572·14-s + 0.258·15-s + 0.250·16-s − 0.742·17-s + 0.235·18-s − 1.22·19-s + 0.223·20-s − 0.467·21-s − 0.973·22-s + 0.340·23-s + 0.204·24-s + 0.200·25-s + 0.317·26-s + 0.192·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| 37 | \( 1 + 37T \) |
| good | 7 | \( 1 + 15T + 343T^{2} \) |
| 11 | \( 1 + 50.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 92.4T + 2.97e4T^{2} \) |
| 41 | \( 1 - 418.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 10.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 692.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 604.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 256.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 385.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 102.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 131.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 190.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3T + 9.12e5T^{2} \) |
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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
−9.067390319827868482280830045114, −8.215241971356809728310252408823, −7.27534755602640002920595316717, −6.43016238123071192543092571516, −5.64883855075278718172805936018, −4.65056697063125303034925956830, −3.62312777389315297434712970495, −2.71833802875145940837054996106, −1.88738428039713629892708528973, 0,
1.88738428039713629892708528973, 2.71833802875145940837054996106, 3.62312777389315297434712970495, 4.65056697063125303034925956830, 5.64883855075278718172805936018, 6.43016238123071192543092571516, 7.27534755602640002920595316717, 8.215241971356809728310252408823, 9.067390319827868482280830045114
