| L(s) = 1 | + (0.959 + 0.281i)2-s + (0.932 − 1.07i)3-s + (0.841 + 0.540i)4-s + (−0.461 + 3.21i)5-s + (1.19 − 0.770i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (0.138 + 0.961i)9-s + (−1.34 + 2.95i)10-s + (−1.39 + 0.409i)11-s + (1.36 − 0.401i)12-s + (1.03 − 2.25i)13-s + (0.142 + 0.989i)14-s + (3.02 + 3.49i)15-s + (0.415 + 0.909i)16-s + (5.39 − 3.46i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.538 − 0.621i)3-s + (0.420 + 0.270i)4-s + (−0.206 + 1.43i)5-s + (0.489 − 0.314i)6-s + (0.157 + 0.343i)7-s + (0.231 + 0.267i)8-s + (0.0460 + 0.320i)9-s + (−0.426 + 0.933i)10-s + (−0.420 + 0.123i)11-s + (0.394 − 0.115i)12-s + (0.286 − 0.626i)13-s + (0.0380 + 0.264i)14-s + (0.781 + 0.902i)15-s + (0.103 + 0.227i)16-s + (1.30 − 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08019 + 0.688546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08019 + 0.688546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.77 + 3.91i)T \) |
| good | 3 | \( 1 + (-0.932 + 1.07i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.461 - 3.21i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (1.39 - 0.409i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.03 + 2.25i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.39 + 3.46i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (6.93 + 4.45i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.68 + 4.29i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.05 + 1.21i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.564 - 3.92i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.50 + 10.4i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (6.54 - 7.55i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 + (-4.96 - 10.8i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.22 - 4.86i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (0.106 + 0.123i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (6.19 + 1.81i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.65 - 2.24i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (5.53 + 3.55i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.403 - 0.882i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.58 + 11.0i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.67 - 8.86i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (0.0665 - 0.463i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.85560941266048695401495215798, −10.81070722762691667934377596777, −10.20746458982431402624189729017, −8.479573074933344985532068686481, −7.72212527732829934609681552115, −6.92443810592464866648600872623, −6.00483856491720570812118495175, −4.62690069488563276202174466305, −3.02547483809233763357619628333, −2.42728483975736700213920330719,
1.52466513153414538758109684714, 3.55587136373157513630029210687, 4.23374615209447542584408362421, 5.25019985072906863862809554138, 6.41108784492574401691615821290, 8.079092373163255506009527506288, 8.602437851624210870316135282946, 9.781596644730871374950396630516, 10.48238548930603924032224951715, 11.81926751095551267212239373698
