| L(s) = 1 | + (0.599 − 0.267i)2-s + (−1.87 + 0.397i)3-s + (−1.04 + 1.16i)4-s + (0.373 + 3.54i)5-s + (−1.01 + 0.738i)6-s + (−1.30 + 2.30i)7-s + (−0.724 + 2.22i)8-s + (0.599 − 0.267i)9-s + (1.17 + 2.02i)10-s + (1.5 − 2.59i)12-s + (4.78 + 3.47i)13-s + (−0.169 + 1.72i)14-s + (−2.10 − 6.49i)15-s + (−0.167 − 1.59i)16-s + (−1.51 − 0.673i)17-s + (0.288 − 0.320i)18-s + ⋯ |
| L(s) = 1 | + (0.424 − 0.188i)2-s + (−1.07 + 0.229i)3-s + (−0.524 + 0.582i)4-s + (0.166 + 1.58i)5-s + (−0.414 + 0.301i)6-s + (−0.493 + 0.869i)7-s + (−0.256 + 0.787i)8-s + (0.199 − 0.0890i)9-s + (0.370 + 0.641i)10-s + (0.433 − 0.749i)12-s + (1.32 + 0.963i)13-s + (−0.0452 + 0.462i)14-s + (−0.544 − 1.67i)15-s + (−0.0417 − 0.397i)16-s + (−0.367 − 0.163i)17-s + (0.0679 − 0.0755i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.113501 - 0.845800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113501 - 0.845800i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.30 - 2.30i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.599 + 0.267i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (1.87 - 0.397i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.373 - 3.54i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-4.78 - 3.47i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.51 + 0.673i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.991 - 1.10i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.951 - 2.92i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.740 + 7.04i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-4.41 - 0.937i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (0.397 - 1.22i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + (1.10 + 1.23i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (0.964 - 9.17i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.92 + 6.57i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.698 - 6.65i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.97 + 5.06i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 3.38i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (5.84 - 2.60i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.135 - 0.0986i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.87 + 5.72i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.99292607419423002273437939161, −9.930359286119689105425066004949, −9.107251988079770910695003608183, −8.109225735105164617399056471349, −6.88749615811924542112664791337, −6.12943813048688113330148999624, −5.58439441255993259427247494222, −4.27457020872420643115793643746, −3.34976441172391501240927409046, −2.39035028106946854137497400335,
0.50713286913173710871916808786, 1.13646005742588839181092312339, 3.66050952441541699924556120948, 4.60940658627693094696387804931, 5.32071617136860068799785073011, 6.04048200396991134543936011170, 6.69541063709480404796067223137, 8.195408667396448848208593837890, 8.870802104601191791346722547714, 9.828001661369048299856198940457
