| L(s) = 1 | + (−0.998 + 0.0475i)2-s + (1.03 − 1.38i)3-s + (0.995 − 0.0950i)4-s + (−2.69 − 2.56i)5-s + (−0.969 + 1.43i)6-s + (0.311 + 1.61i)7-s + (−0.989 + 0.142i)8-s + (−0.851 − 2.87i)9-s + (2.81 + 2.43i)10-s + (0.912 − 0.470i)11-s + (0.899 − 1.48i)12-s + (0.754 + 0.145i)13-s + (−0.387 − 1.59i)14-s + (−6.34 + 1.07i)15-s + (0.981 − 0.189i)16-s + (−2.48 − 5.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.706 + 0.0336i)2-s + (0.598 − 0.801i)3-s + (0.497 − 0.0475i)4-s + (−1.20 − 1.14i)5-s + (−0.395 + 0.586i)6-s + (0.117 + 0.609i)7-s + (−0.349 + 0.0503i)8-s + (−0.283 − 0.958i)9-s + (0.888 + 0.770i)10-s + (0.275 − 0.141i)11-s + (0.259 − 0.427i)12-s + (0.209 + 0.0403i)13-s + (−0.103 − 0.426i)14-s + (−1.63 + 0.277i)15-s + (0.245 − 0.0473i)16-s + (−0.603 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.138288 - 0.666381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.138288 - 0.666381i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.998 - 0.0475i)T \) |
| 3 | \( 1 + (-1.03 + 1.38i)T \) |
| 23 | \( 1 + (1.38 - 4.59i)T \) |
| good | 5 | \( 1 + (2.69 + 2.56i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.311 - 1.61i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-0.912 + 0.470i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-0.754 - 0.145i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (2.48 + 5.45i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (5.96 + 2.72i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.130 - 1.36i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (2.08 - 1.64i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-2.95 + 10.0i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.602 - 0.631i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (5.94 - 7.56i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-4.14 - 2.39i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.30 + 1.50i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.22 + 11.5i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-2.30 + 5.76i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-5.12 + 9.94i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-0.429 - 0.667i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.14 + 2.51i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-15.1 + 5.24i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-3.86 + 3.68i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-0.920 + 6.39i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.06 + 0.257i)T + (86.2 + 44.4i)T^{2} \) |
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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
−11.15585372082305232053119424960, −9.275168649506842470041477885834, −8.982360623394203897828779976321, −8.145521635872117332830574516263, −7.44529966629588159398853115417, −6.39848277893420238694630506911, −4.94913158108242053498491121994, −3.60633042151701140362710784342, −2.08775540264163348373200462457, −0.49780266360336015312511324695,
2.34629055352117062683251314134, 3.73775205715432504712270802292, 4.22579574454332755707293932288, 6.27866986212578208953611502103, 7.19282204823259837500824949966, 8.211273649702335183796602520302, 8.598059460709347755953056732852, 10.13688340611703633474576195496, 10.54757131278742346495579777822, 11.16063653793491924753987581013
