| L(s) = 1 | + (−5.56 − 24.3i)2-s + (167. + 155. i)3-s + (−101. + 48.7i)4-s + (−1.83e3 + 276. i)5-s + (2.84e3 − 4.93e3i)6-s + (−303. − 525. i)7-s + (−6.22e3 − 7.80e3i)8-s + (2.41e3 + 3.22e4i)9-s + (1.69e4 + 4.31e4i)10-s + (2.38e4 + 1.14e4i)11-s + (−2.44e4 − 7.55e3i)12-s + (−6.24e3 + 1.59e4i)13-s + (−1.11e4 + 1.03e4i)14-s + (−3.49e5 − 2.38e5i)15-s + (−1.91e5 + 2.40e5i)16-s + (−6.46e5 − 9.74e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.245 − 1.07i)2-s + (1.19 + 1.10i)3-s + (−0.197 + 0.0952i)4-s + (−1.31 + 0.198i)5-s + (0.897 − 1.55i)6-s + (−0.0477 − 0.0827i)7-s + (−0.537 − 0.673i)8-s + (0.122 + 1.63i)9-s + (0.536 + 1.36i)10-s + (0.490 + 0.236i)11-s + (−0.341 − 0.105i)12-s + (−0.0606 + 0.154i)13-s + (−0.0773 + 0.0717i)14-s + (−1.78 − 1.21i)15-s + (−0.730 + 0.915i)16-s + (−1.87 − 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.100262 + 0.258256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100262 + 0.258256i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.30e7 - 1.82e7i)T \) |
| good | 2 | \( 1 + (5.56 + 24.3i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-167. - 155. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (1.83e3 - 276. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (303. + 525. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.38e4 - 1.14e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (6.24e3 - 1.59e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (6.46e5 + 9.74e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-3.20e4 + 4.28e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (1.36e6 - 9.27e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (4.75e6 - 4.41e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (4.33e6 + 1.33e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (-5.86e6 + 1.01e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.59e6 - 6.96e6i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-1.17e7 + 5.66e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-1.13e7 - 2.89e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-3.37e7 + 4.23e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-3.05e7 + 9.42e6i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (1.54e7 - 2.06e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (-8.72e7 - 5.94e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (2.74e6 - 6.99e6i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (2.89e8 + 5.01e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-5.65e8 - 5.25e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (6.37e8 + 5.91e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-1.78e8 - 8.57e7i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.69189830165045588245363164836, −13.14807867712770540857261979588, −11.58037354602905304878208985656, −10.90443809207628424867602858289, −9.546964853344909611615896590768, −8.770921767818614889476453587479, −7.19712860493446593134874592890, −4.25483599652362316477470929400, −3.53875285407249694891010705603, −2.21153136265280400716256042823,
0.082103157493033422332997564275, 2.19626444314916565120881328717, 3.88014491190457979488264012599, 6.34596403293229502278579701338, 7.42255890313444018352176050601, 8.193616723821301612906547438263, 8.929009756293684358275475541432, 11.47121999623789423481342786230, 12.50913472979005104652893829293, 13.79584000507821627997393554511
