| L(s) = 1 | + (−1.29 + 0.0619i)2-s + (1.73 + 0.0507i)3-s + (−0.305 + 0.0291i)4-s + (−1.59 − 1.51i)5-s + (−2.25 + 0.0411i)6-s + (0.597 + 3.09i)7-s + (2.97 − 0.427i)8-s + (2.99 + 0.175i)9-s + (2.16 + 1.87i)10-s + (5.77 − 2.97i)11-s + (−0.529 + 0.0349i)12-s + (0.586 + 0.113i)13-s + (−0.968 − 3.99i)14-s + (−2.68 − 2.71i)15-s + (−3.23 + 0.623i)16-s + (0.498 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 + 0.0437i)2-s + (0.999 + 0.0293i)3-s + (−0.152 + 0.0145i)4-s + (−0.712 − 0.679i)5-s + (−0.920 + 0.0168i)6-s + (0.225 + 1.17i)7-s + (1.05 − 0.151i)8-s + (0.998 + 0.0586i)9-s + (0.684 + 0.593i)10-s + (1.73 − 0.896i)11-s + (−0.152 + 0.0100i)12-s + (0.162 + 0.0313i)13-s + (−0.258 − 1.06i)14-s + (−0.692 − 0.699i)15-s + (−0.808 + 0.155i)16-s + (0.120 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.964262 + 0.00541246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.964262 + 0.00541246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 - 0.0507i)T \) |
| 23 | \( 1 + (4.43 + 1.83i)T \) |
| good | 2 | \( 1 + (1.29 - 0.0619i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (1.59 + 1.51i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.597 - 3.09i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-5.77 + 2.97i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-0.586 - 0.113i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.498 - 1.09i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.29 - 1.04i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.128 - 1.34i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-3.48 + 2.74i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (1.94 - 6.63i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (7.43 - 7.79i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (1.53 - 1.95i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (3.46 + 2.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.44 + 7.44i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.762 + 3.95i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (0.785 - 1.96i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (3.18 - 6.18i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (5.22 + 8.13i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.46 + 14.1i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.600 - 0.207i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (4.89 - 4.66i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (0.587 - 4.08i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (18.6 + 4.51i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.22883561674461233117542386551, −11.55162292134990362993216821649, −9.949298769035097586895622665148, −9.150510276537252551725876883806, −8.392829602564126486162997610949, −8.118411115202820643660924961574, −6.46335673442463834153746336879, −4.66918371863047338741908613221, −3.55136523961876222185760958100, −1.46643583318201835600420497830,
1.47846563222599345707770276447, 3.67263389954670369343544790428, 4.36780125544873040874506129941, 7.05201551911162768466040657119, 7.35218182681346407440459398951, 8.447819851684226306540423298057, 9.464527296254764879134655794013, 10.12489969966597412345277479282, 11.16066035947997214579286836848, 12.28467292358416436760905314314
