| 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
−12.03908440247638327145198069130, −11.41144792858009977320991878592, −10.39795494926652981649884527311, −9.052202185574262784495743455507, −8.552669129858716649157066850332, −7.75390593247524494722352289778, −6.44228650994076984635697712690, −5.81152289217214344964101228146, −2.65506285630339289342909868701, −1.49605198743694194713535155775,
0.992619150927682938121678904987, 3.26219012043716451709111196804, 4.82757769882967377370173113907, 6.59614675455193304708908508787, 7.64059963459005921356075827439, 8.579299804020469830137333212445, 9.589656238571813828250050492844, 10.18409811261686340823422681542, 11.02210549328335695648503347090, 11.58522480749470007358243074919
