| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 2.62·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 5.24·10-s + 4.49·11-s − 12·12-s − 13·13-s − 14·14-s + 7.87·15-s + 16·16-s + 12.2·17-s + 18·18-s + 62.1·19-s − 10.4·20-s + 21·21-s + 8.98·22-s − 58.1·23-s − 24·24-s − 118.·25-s − 26·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.234·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.165·10-s + 0.123·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.135·15-s + 0.250·16-s + 0.174·17-s + 0.235·18-s + 0.749·19-s − 0.117·20-s + 0.218·21-s + 0.0871·22-s − 0.526·23-s − 0.204·24-s − 0.944·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 2.62T + 125T^{2} \) |
| 11 | \( 1 - 4.49T + 1.33e3T^{2} \) |
| 17 | \( 1 - 12.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 182.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 280.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 151.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 55.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 147.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 652.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 546.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 715.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 60.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 817.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 616.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−10.05077144647553301172166080050, −9.283496287308514536436207526407, −7.83680143586349202506360944358, −7.13508038206158898621610989809, −6.04987416309986973573535021655, −5.35658726040189688495352687733, −4.20574355703643239077945068406, −3.27226146622143242476150069206, −1.74458790150880540260686909809, 0,
1.74458790150880540260686909809, 3.27226146622143242476150069206, 4.20574355703643239077945068406, 5.35658726040189688495352687733, 6.04987416309986973573535021655, 7.13508038206158898621610989809, 7.83680143586349202506360944358, 9.283496287308514536436207526407, 10.05077144647553301172166080050
