| L(s) = 1 | − 2·2-s − 1.93·3-s + 4·4-s − 5·5-s + 3.87·6-s + 17.2·7-s − 8·8-s − 23.2·9-s + 10·10-s − 11·11-s − 7.75·12-s − 14.1·13-s − 34.4·14-s + 9.69·15-s + 16·16-s + 17·17-s + 46.4·18-s − 21.4·19-s − 20·20-s − 33.3·21-s + 22·22-s − 33.4·23-s + 15.5·24-s + 25·25-s + 28.2·26-s + 97.3·27-s + 68.8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.373·3-s + 0.5·4-s − 0.447·5-s + 0.263·6-s + 0.928·7-s − 0.353·8-s − 0.860·9-s + 0.316·10-s − 0.301·11-s − 0.186·12-s − 0.301·13-s − 0.656·14-s + 0.166·15-s + 0.250·16-s + 0.242·17-s + 0.608·18-s − 0.258·19-s − 0.223·20-s − 0.346·21-s + 0.213·22-s − 0.303·23-s + 0.131·24-s + 0.200·25-s + 0.213·26-s + 0.694·27-s + 0.464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 1.93T + 27T^{2} \) |
| 7 | \( 1 - 17.2T + 343T^{2} \) |
| 13 | \( 1 + 14.1T + 2.19e3T^{2} \) |
| 19 | \( 1 + 21.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 342.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 151.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 377.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 23.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 481.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 778.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 500.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 823.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 308.T + 9.12e5T^{2} \) |
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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
−8.388796042470440380466405130661, −7.940407043185735047675562227315, −7.02226618218575073601651362346, −6.17460148055309265489640874738, −5.24401335958892771160003198955, −4.53005877178003678935679909245, −3.21838934863523589871895751382, −2.27494019771148918764190239670, −1.04877897056671794919442263584, 0,
1.04877897056671794919442263584, 2.27494019771148918764190239670, 3.21838934863523589871895751382, 4.53005877178003678935679909245, 5.24401335958892771160003198955, 6.17460148055309265489640874738, 7.02226618218575073601651362346, 7.940407043185735047675562227315, 8.388796042470440380466405130661
