| L(s) = 1 | + 1.88·2-s + 3-s + 1.56·4-s − 1.95·5-s + 1.88·6-s − 1.51·7-s − 0.819·8-s + 9-s − 3.68·10-s + 1.07·11-s + 1.56·12-s + 4.52·13-s − 2.85·14-s − 1.95·15-s − 4.67·16-s − 17-s + 1.88·18-s + 5.39·19-s − 3.05·20-s − 1.51·21-s + 2.02·22-s + 4.97·23-s − 0.819·24-s − 1.19·25-s + 8.54·26-s + 27-s − 2.36·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.782·4-s − 0.872·5-s + 0.770·6-s − 0.571·7-s − 0.289·8-s + 0.333·9-s − 1.16·10-s + 0.323·11-s + 0.452·12-s + 1.25·13-s − 0.762·14-s − 0.503·15-s − 1.16·16-s − 0.242·17-s + 0.445·18-s + 1.23·19-s − 0.683·20-s − 0.329·21-s + 0.431·22-s + 1.03·23-s − 0.167·24-s − 0.238·25-s + 1.67·26-s + 0.192·27-s − 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.959476965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.959476965\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 5.29T + 73T^{2} \) |
| 83 | \( 1 + 6.35T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−8.334180291709211456879727351238, −7.60844334080071241804568184701, −6.85298013944806139821395724274, −6.10821240640600086384574947932, −5.40386161105101436462964253461, −4.38944394749907836014572359228, −3.80584221272847528080951077895, −3.32251105009986082354094975042, −2.48880509843907235141206550517, −0.926906881461073500932761636345,
0.926906881461073500932761636345, 2.48880509843907235141206550517, 3.32251105009986082354094975042, 3.80584221272847528080951077895, 4.38944394749907836014572359228, 5.40386161105101436462964253461, 6.10821240640600086384574947932, 6.85298013944806139821395724274, 7.60844334080071241804568184701, 8.334180291709211456879727351238
