| L(s) = 1 | + (−0.565 − 1.29i)2-s + (−1.67 − 0.457i)3-s + (−1.36 + 1.46i)4-s + (1.52 + 1.81i)5-s + (0.350 + 2.42i)6-s + (0.767 − 0.279i)7-s + (2.66 + 0.934i)8-s + (2.58 + 1.52i)9-s + (1.49 − 3.00i)10-s + (3.58 − 4.26i)11-s + (2.94 − 1.82i)12-s + (−2.36 + 0.417i)13-s + (−0.795 − 0.836i)14-s + (−1.71 − 3.72i)15-s + (−0.297 − 3.98i)16-s + (1.52 + 2.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.399 − 0.916i)2-s + (−0.964 − 0.264i)3-s + (−0.680 + 0.732i)4-s + (0.681 + 0.811i)5-s + (0.143 + 0.989i)6-s + (0.289 − 0.105i)7-s + (0.943 + 0.330i)8-s + (0.860 + 0.509i)9-s + (0.471 − 0.948i)10-s + (1.08 − 1.28i)11-s + (0.849 − 0.526i)12-s + (−0.656 + 0.115i)13-s + (−0.212 − 0.223i)14-s + (−0.442 − 0.962i)15-s + (−0.0743 − 0.997i)16-s + (0.370 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.774679 - 0.350020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.774679 - 0.350020i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.565 + 1.29i)T \) |
| 3 | \( 1 + (1.67 + 0.457i)T \) |
| good | 5 | \( 1 + (-1.52 - 1.81i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.767 + 0.279i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-3.58 + 4.26i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.36 - 0.417i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 2.64i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.631 + 0.364i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.38 - 2.32i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-9.24 - 1.62i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.92 - 0.701i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-4.61 + 2.66i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.29 + 7.32i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.42 - 5.26i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 3.72i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 2.99iT - 53T^{2} \) |
| 59 | \( 1 + (-0.837 - 0.998i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.528 + 1.45i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.70 - 0.829i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.37 + 7.57i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.62 - 4.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.739 - 4.19i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (14.1 + 2.49i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.96 + 8.60i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.8 + 9.96i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.90070366777068111718458264298, −11.16404910031640662468899074012, −10.50436538679001063060366540386, −9.593922771598354466507776103984, −8.366431529621388308017556593100, −7.01727859503417485466240706247, −6.04705255683562383419076889040, −4.64091780000674740508804056306, −3.04831698734666260214301622360, −1.32214957660741767448731651577,
1.29204358081516444623974145285, 4.67761422439814444194094960142, 4.99741519101466295133900282473, 6.36774634224102389409325114532, 7.12685044018791372280313507038, 8.584295444256513387719995993380, 9.702402795300994545503017161528, 9.980911766585467210720719487467, 11.53050976388364090346381685368, 12.46205097693572764010991311280
