| L(s) = 1 | + (−2.28 + 0.108i)2-s + (1.58 − 0.702i)3-s + (3.23 − 0.308i)4-s + (−0.595 − 0.568i)5-s + (−3.54 + 1.77i)6-s + (−0.645 − 3.34i)7-s + (−2.82 + 0.405i)8-s + (2.01 − 2.22i)9-s + (1.42 + 1.23i)10-s + (−3.24 + 1.67i)11-s + (4.89 − 2.75i)12-s + (1.19 + 0.231i)13-s + (1.84 + 7.58i)14-s + (−1.34 − 0.481i)15-s + (0.0407 − 0.00785i)16-s + (−1.97 − 4.32i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 + 0.0770i)2-s + (0.914 − 0.405i)3-s + (1.61 − 0.154i)4-s + (−0.266 − 0.254i)5-s + (−1.44 + 0.726i)6-s + (−0.243 − 1.26i)7-s + (−0.998 + 0.143i)8-s + (0.671 − 0.741i)9-s + (0.450 + 0.390i)10-s + (−0.978 + 0.504i)11-s + (1.41 − 0.796i)12-s + (0.332 + 0.0641i)13-s + (0.492 + 2.02i)14-s + (−0.346 − 0.124i)15-s + (0.0101 − 0.00196i)16-s + (−0.479 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.467195 - 0.452282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.467195 - 0.452282i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.58 + 0.702i)T \) |
| 23 | \( 1 + (-4.57 + 1.44i)T \) |
| good | 2 | \( 1 + (2.28 - 0.108i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (0.595 + 0.568i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.645 + 3.34i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (3.24 - 1.67i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-1.19 - 0.231i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (1.97 + 4.32i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.51 - 0.689i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.595 + 6.23i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (1.18 - 0.929i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (2.51 - 8.55i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.38 - 1.45i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-1.04 + 1.32i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-9.52 - 5.50i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.16 - 4.80i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.504 + 2.61i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (4.17 - 10.4i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (3.30 - 6.40i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-6.72 - 10.4i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.56 + 12.1i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.56 + 2.96i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (4.58 - 4.36i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (0.898 - 6.24i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-13.9 - 3.38i)T + (86.2 + 44.4i)T^{2} \) |
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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
−11.99220354929202413831994537222, −10.70425851009457926579518470162, −10.00158008011127516486240212822, −9.122269989466294396966354537969, −8.138407454986953800547661068435, −7.43815861900613071204164388548, −6.74922086519458379663759636272, −4.38764779604688639698534148498, −2.63281929800381546449966487114, −0.865208629179986896754823279397,
2.12183753218060239963901181516, 3.31139532562667665113589792210, 5.41870035887508460027331264384, 7.05422995789548423908448173181, 8.054172348280873160915772806375, 8.835936374589893930321197308445, 9.315463412907135770459273880649, 10.60019658657496335316485267356, 11.04150141835520875979940436620, 12.51927734843198385154809854901
