| L(s) = 1 | + 1.70·2-s + 3-s + 0.896·4-s + 1.19·5-s + 1.70·6-s + 0.556·7-s − 1.87·8-s + 9-s + 2.04·10-s − 5.84·11-s + 0.896·12-s + 1.11·13-s + 0.947·14-s + 1.19·15-s − 4.98·16-s − 17-s + 1.70·18-s + 5.86·19-s + 1.07·20-s + 0.556·21-s − 9.94·22-s − 8.26·23-s − 1.87·24-s − 3.56·25-s + 1.90·26-s + 27-s + 0.499·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.577·3-s + 0.448·4-s + 0.536·5-s + 0.694·6-s + 0.210·7-s − 0.663·8-s + 0.333·9-s + 0.645·10-s − 1.76·11-s + 0.258·12-s + 0.310·13-s + 0.253·14-s + 0.309·15-s − 1.24·16-s − 0.242·17-s + 0.401·18-s + 1.34·19-s + 0.240·20-s + 0.121·21-s − 2.12·22-s − 1.72·23-s − 0.383·24-s − 0.712·25-s + 0.373·26-s + 0.192·27-s + 0.0943·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 - 0.556T + 7T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 19 | \( 1 - 5.86T + 19T^{2} \) |
| 23 | \( 1 + 8.26T + 23T^{2} \) |
| 29 | \( 1 + 0.910T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 6.72T + 37T^{2} \) |
| 41 | \( 1 + 0.589T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 + 9.83T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 - 3.63T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 9.98T + 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−7.62208126374198313761501201799, −6.49834756784865157938460830518, −5.97264028302790457049556226053, −5.16948746700882164707584848973, −4.84779151265482076653003610289, −3.86024886680050282842660463421, −3.16749040202264973018365576979, −2.51704637052828820834291999184, −1.70717785744540917349280316656, 0,
1.70717785744540917349280316656, 2.51704637052828820834291999184, 3.16749040202264973018365576979, 3.86024886680050282842660463421, 4.84779151265482076653003610289, 5.16948746700882164707584848973, 5.97264028302790457049556226053, 6.49834756784865157938460830518, 7.62208126374198313761501201799
