| L(s) = 1 | + 21.0·2-s + (0.504 + 0.873i)3-s + 316.·4-s + (−262. − 454. i)5-s + (10.6 + 18.4i)6-s + (58.1 − 100. i)7-s + 3.97e3·8-s + (1.09e3 − 1.89e3i)9-s + (−5.53e3 − 9.59e3i)10-s + 4.73e3·11-s + (159. + 276. i)12-s + (−5.48e3 + 9.49e3i)13-s + (1.22e3 − 2.12e3i)14-s + (265. − 459. i)15-s + 4.33e4·16-s + (−5.78e3 + 1.00e4i)17-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + (0.0107 + 0.0186i)3-s + 2.47·4-s + (−0.939 − 1.62i)5-s + (0.0201 + 0.0348i)6-s + (0.0640 − 0.111i)7-s + 2.74·8-s + (0.499 − 0.865i)9-s + (−1.75 − 3.03i)10-s + 1.07·11-s + (0.0266 + 0.0462i)12-s + (−0.691 + 1.19i)13-s + (0.119 − 0.206i)14-s + (0.0202 − 0.0351i)15-s + 2.64·16-s + (−0.285 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.50371 - 2.12455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.50371 - 2.12455i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.76e4 - 5.20e5i)T \) |
| good | 2 | \( 1 - 21.0T + 128T^{2} \) |
| 3 | \( 1 + (-0.504 - 0.873i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (262. + 454. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-58.1 + 100. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 4.73e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.48e3 - 9.49e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (5.78e3 - 1.00e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.11e4 - 3.66e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.05e4 + 1.82e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.76e4 + 8.25e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.30e5 + 2.25e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.00e4 - 3.46e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 4.07e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 8.57e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-6.02e5 - 1.04e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 2.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (5.27e5 - 9.13e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (5.54e5 + 9.59e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (9.78e5 - 1.69e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.52e6 - 2.64e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-4.15e6 + 7.20e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.84e5 - 3.18e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.40e6 + 2.43e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 7.03e6T + 8.07e13T^{2} \) |
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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
−14.25471406025786646815923331742, −12.92491869540621971456429987037, −12.09798658572424621499619552799, −11.66449311506216727209384097255, −9.321426583109521364148370895428, −7.52152774869298321612926954711, −6.04004686414462067832627633453, −4.36331813870041042249001591441, −4.00300298346375207018670953958, −1.41018990250302125552986787671,
2.57530653128205763711169280395, 3.62925527089696195683765130045, 5.08238244700478945971668792524, 6.83218027095815629641358398708, 7.43352501610720068760104231174, 10.52082825157750472409403127631, 11.34300180542603506232140281772, 12.29939912818315807555598501500, 13.68368132809756212711068209449, 14.54142324605400302421267958723
