L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (1.62 + 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.173 − 0.300i)11-s + (−0.642 + 0.766i)13-s + (−1.76 − 0.642i)14-s + (0.766 − 0.642i)16-s − 0.999i·18-s + (−0.173 − 0.984i)19-s + (0.223 + 0.266i)22-s + (0.524 + 1.43i)23-s + (0.5 − 0.866i)26-s + (1.85 + 0.326i)28-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (1.62 + 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.173 − 0.300i)11-s + (−0.642 + 0.766i)13-s + (−1.76 − 0.642i)14-s + (0.766 − 0.642i)16-s − 0.999i·18-s + (−0.173 − 0.984i)19-s + (0.223 + 0.266i)22-s + (0.524 + 1.43i)23-s + (0.5 − 0.866i)26-s + (1.85 + 0.326i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8933282714\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8933282714\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
good | 3 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.642 - 0.766i)T + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53iT - T^{2} \) |
| 41 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.642 + 1.76i)T + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
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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
−8.678663448361280706779055448461, −8.257401171450184778415097758016, −7.57374542284768045247082526841, −6.94864075978801604928296500191, −5.86647156722374131847995702601, −5.11498510415167895927423816591, −4.71894595037190342882151347244, −3.00162280941763333422372465494, −2.14909085073748212892311383992, −1.52863128661068222437468926906,
0.71757907700913506663827882447, 1.70870756343875180248521877538, 2.72987191075976230328357279156, 3.80681252109303729416813572369, 4.61091005793972721232595373415, 5.59040418580738824131837130038, 6.52261332564005836742053933288, 7.30348844219780215429303656719, 7.87201553074766484168904335566, 8.368486478316412973847635814496