| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.380002 + 0.0964267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380002 + 0.0964267i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.13282342847608249708004627074, −9.325789738617707367088314087623, −8.786562482613047057978876602622, −8.024096919699505622189432849841, −6.58787926131947498907919294833, −5.54272029187028502310649915629, −5.08512072430864547972307507688, −4.50722846142026306349138394582, −2.10144812588059421045567911526, −1.04306437223112086238794089964,
0.31467881301328820645297472807, 2.63300901393478152671824496974, 3.78098159088674569761399906244, 4.80013309957531135658876286822, 5.83242354871717611819876156518, 6.82486173207058645320893860754, 7.56463218556793417721775342989, 8.072610857549563466110337432259, 9.774478709499415703961491290904, 10.03964549027152840868851047738
