| L(s) = 1 | + (−0.110 − 0.223i)3-s + (−0.800 + 0.701i)5-s + (−1.28 + 2.31i)7-s + (1.78 − 2.33i)9-s + (3.58 − 0.235i)11-s + (1.36 + 1.36i)13-s + (0.245 + 0.101i)15-s + (−3.82 + 1.54i)17-s + (0.937 + 7.11i)19-s + (0.659 + 0.0324i)21-s + (7.25 + 3.57i)23-s + (−0.504 + 3.83i)25-s + (−1.45 − 0.289i)27-s + (0.397 + 1.99i)29-s + (5.57 − 2.75i)31-s + ⋯ |
| L(s) = 1 | + (−0.0637 − 0.129i)3-s + (−0.357 + 0.313i)5-s + (−0.485 + 0.874i)7-s + (0.596 − 0.776i)9-s + (1.08 − 0.0709i)11-s + (0.378 + 0.378i)13-s + (0.0633 + 0.0262i)15-s + (−0.927 + 0.373i)17-s + (0.215 + 1.63i)19-s + (0.143 + 0.00708i)21-s + (1.51 + 0.745i)23-s + (−0.100 + 0.766i)25-s + (−0.279 − 0.0556i)27-s + (0.0737 + 0.370i)29-s + (1.00 − 0.494i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20903 + 0.517610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20903 + 0.517610i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.28 - 2.31i)T \) |
| 17 | \( 1 + (3.82 - 1.54i)T \) |
| good | 3 | \( 1 + (0.110 + 0.223i)T + (-1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (0.800 - 0.701i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (-3.58 + 0.235i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 1.36i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.937 - 7.11i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-7.25 - 3.57i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.397 - 1.99i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-5.57 + 2.75i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.117 + 1.78i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-0.719 + 3.61i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.66 + 6.42i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (0.501 - 1.87i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.76 + 3.60i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-9.46 - 1.24i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-3.71 - 10.9i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (13.7 + 7.94i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.66 - 5.11i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (11.9 + 4.06i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (0.276 - 0.560i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-2.77 + 6.70i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (15.6 + 4.19i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (6.30 - 1.25i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.36964466797167433095669935590, −10.13675687536522977744650394208, −9.224644086938623996016274237865, −8.660900161596278919535925353365, −7.25452886391513091772533666873, −6.53384471540688302335839779817, −5.69260425735926310457065894642, −4.10482578954709387496280319227, −3.30446573680036534102643754024, −1.55840516018418995849788571932,
0.932144614813464166262289846418, 2.87053497384524878004656492189, 4.31521425242726446283332176463, 4.79022961757121759355867912279, 6.56628798445918636135106444522, 7.02362326169824470459969428249, 8.227821631814940014477524622250, 9.129327294997180727198497335192, 10.03158132459443081143344531259, 10.95165687055840715105450630566
