| L(s) = 1 | + (−0.796 + 0.0379i)2-s + (−1.59 − 0.677i)3-s + (−1.35 + 0.129i)4-s + (0.279 + 0.266i)5-s + (1.29 + 0.479i)6-s + (−0.0444 − 0.230i)7-s + (2.65 − 0.381i)8-s + (2.08 + 2.16i)9-s + (−0.233 − 0.201i)10-s + (−3.40 + 1.75i)11-s + (2.25 + 0.713i)12-s + (6.25 + 1.20i)13-s + (0.0441 + 0.182i)14-s + (−0.265 − 0.614i)15-s + (0.576 − 0.111i)16-s + (1.94 + 4.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.563 + 0.0268i)2-s + (−0.920 − 0.391i)3-s + (−0.678 + 0.0648i)4-s + (0.125 + 0.119i)5-s + (0.529 + 0.195i)6-s + (−0.0168 − 0.0871i)7-s + (0.939 − 0.135i)8-s + (0.693 + 0.720i)9-s + (−0.0737 − 0.0638i)10-s + (−1.02 + 0.529i)11-s + (0.649 + 0.205i)12-s + (1.73 + 0.334i)13-s + (0.0118 + 0.0486i)14-s + (−0.0684 − 0.158i)15-s + (0.144 − 0.0277i)16-s + (0.471 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.532346 + 0.194874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.532346 + 0.194874i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.59 + 0.677i)T \) |
| 23 | \( 1 + (2.41 - 4.14i)T \) |
| good | 2 | \( 1 + (0.796 - 0.0379i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (-0.279 - 0.266i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.0444 + 0.230i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (3.40 - 1.75i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-6.25 - 1.20i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.94 - 4.25i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-4.94 - 2.25i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.111 + 1.16i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-0.690 + 0.543i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (1.01 - 3.47i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.33 - 1.39i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (6.32 - 8.04i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-5.95 - 3.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.729 + 0.842i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.78 + 9.28i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (2.28 - 5.69i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-2.98 + 5.79i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-6.03 - 9.38i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.69 - 5.90i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-13.6 + 4.72i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (8.54 - 8.14i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-1.66 + 11.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.06 + 0.259i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.54755875953981705389526284811, −11.42087783047735594802252831368, −10.40014612152126036555026275803, −9.831351220929705105313243724183, −8.322380811210089730861840198897, −7.67061137872352838445640341341, −6.26730190207638432421525732814, −5.26787209836664004527276958101, −3.92247237488302851405736876409, −1.39017261168643784027581020136,
0.799213469238351928127900293717, 3.58366513543653193538385384664, 5.08242440077620305171152065680, 5.73159252476403396813955694132, 7.30722762256310904039813267340, 8.526420754365044673298846470823, 9.384059260687267448544175818756, 10.43031337941762985961893815834, 10.99668454375390583465749588630, 12.14783696737799009820515324914
