| L(s) = 1 | + (−2.49 − 0.119i)2-s + (−0.0685 + 1.73i)3-s + (4.23 + 0.404i)4-s + (0.378 − 0.361i)5-s + (0.377 − 4.31i)6-s + (0.824 − 4.27i)7-s + (−5.58 − 0.802i)8-s + (−2.99 − 0.237i)9-s + (−0.988 + 0.856i)10-s + (0.529 + 0.273i)11-s + (−0.990 + 7.30i)12-s + (2.87 − 0.554i)13-s + (−2.56 + 10.5i)14-s + (0.598 + 0.680i)15-s + (5.49 + 1.05i)16-s + (2.39 − 5.24i)17-s + ⋯ |
| L(s) = 1 | + (−1.76 − 0.0841i)2-s + (−0.0395 + 0.999i)3-s + (2.11 + 0.202i)4-s + (0.169 − 0.161i)5-s + (0.153 − 1.76i)6-s + (0.311 − 1.61i)7-s + (−1.97 − 0.283i)8-s + (−0.996 − 0.0790i)9-s + (−0.312 + 0.270i)10-s + (0.159 + 0.0823i)11-s + (−0.285 + 2.10i)12-s + (0.798 − 0.153i)13-s + (−0.686 + 2.82i)14-s + (0.154 + 0.175i)15-s + (1.37 + 0.264i)16-s + (0.580 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563969 - 0.0688519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563969 - 0.0688519i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0685 - 1.73i)T \) |
| 23 | \( 1 + (-4.58 + 1.40i)T \) |
| good | 2 | \( 1 + (2.49 + 0.119i)T + (1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (-0.378 + 0.361i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.824 + 4.27i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (-0.529 - 0.273i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-2.87 + 0.554i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (-2.39 + 5.24i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.361 - 0.165i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.232 + 2.43i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (-5.66 - 4.45i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-0.882 - 3.00i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.98 - 6.27i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (5.40 + 6.87i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (8.38 - 4.83i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.79 + 6.68i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.04 + 5.39i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (0.724 + 1.80i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (6.35 + 12.3i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (2.85 - 4.44i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.48 - 9.82i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (7.16 + 2.47i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (1.29 + 1.23i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-0.859 - 5.97i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-12.9 + 3.15i)T + (86.2 - 44.4i)T^{2} \) |
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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
−11.58522480749470007358243074919, −11.02210549328335695648503347090, −10.18409811261686340823422681542, −9.589656238571813828250050492844, −8.579299804020469830137333212445, −7.64059963459005921356075827439, −6.59614675455193304708908508787, −4.82757769882967377370173113907, −3.26219012043716451709111196804, −0.992619150927682938121678904987,
1.49605198743694194713535155775, 2.65506285630339289342909868701, 5.81152289217214344964101228146, 6.44228650994076984635697712690, 7.75390593247524494722352289778, 8.552669129858716649157066850332, 9.052202185574262784495743455507, 10.39795494926652981649884527311, 11.41144792858009977320991878592, 12.03908440247638327145198069130
