L(s) = 1 | + 9.96·2-s + (−9.83 − 17.0i)3-s + 67.3·4-s + (6.65 + 11.5i)5-s + (−98.0 − 169. i)6-s + (61.0 − 105. i)7-s + 352.·8-s + (−71.8 + 124. i)9-s + (66.3 + 114. i)10-s − 64.2·11-s + (−662. − 1.14e3i)12-s + (38.2 − 66.1i)13-s + (608. − 1.05e3i)14-s + (130. − 226. i)15-s + 1.35e3·16-s + (−984. + 1.70e3i)17-s + ⋯ |
L(s) = 1 | + 1.76·2-s + (−0.630 − 1.09i)3-s + 2.10·4-s + (0.119 + 0.206i)5-s + (−1.11 − 1.92i)6-s + (0.470 − 0.815i)7-s + 1.94·8-s + (−0.295 + 0.512i)9-s + (0.209 + 0.363i)10-s − 0.159·11-s + (−1.32 − 2.29i)12-s + (0.0627 − 0.108i)13-s + (0.829 − 1.43i)14-s + (0.150 − 0.260i)15-s + 1.32·16-s + (−0.825 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.19191 - 1.78158i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.19191 - 1.78158i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-2.28e3 + 1.19e4i)T \) |
good | 2 | \( 1 - 9.96T + 32T^{2} \) |
| 3 | \( 1 + (9.83 + 17.0i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-6.65 - 11.5i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-61.0 + 105. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + 64.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-38.2 + 66.1i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (984. - 1.70e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.24e3 - 2.15e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-2.29e3 - 3.97e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-887. + 1.53e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (721. + 1.24e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (4.28e3 + 7.41e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 4.83e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 9.50e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.02e4 + 1.77e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + 1.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (9.71e3 - 1.68e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.42e4 - 5.93e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.17e4 + 3.76e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.74e4 + 4.74e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.42e4 - 2.47e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.05e3 + 5.29e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (3.71e4 + 6.42e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 6.58e4T + 8.58e9T^{2} \) |
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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
−14.35207983797527134825912415527, −13.44965483850288748248332052973, −12.65817737954645890748793150792, −11.65132780367982573351414597650, −10.63969800360403131257908608928, −7.65200107742957051338031572283, −6.58170512834885446835571330137, −5.49449389775189661171836316468, −3.81109786820895079167602922994, −1.65192217326734522796925418638,
2.83169590593425210603106492518, 4.86512682116329196285544892237, 5.04032606410674356346924377473, 6.75218606536117771741494550931, 9.172520851291382547891807907149, 10.97852163116473352659144683540, 11.57623510745468421283860165739, 12.85251411093755660163144930155, 14.01261707110960304858588711469, 15.25356763495356678483490422555