| L(s) = 1 | + (3.93 + 4.92i)2-s + (−23.5 − 3.55i)3-s + (−1.72 + 7.54i)4-s + (6.81 + 4.64i)5-s + (−75.1 − 130. i)6-s + (57.5 − 99.7i)7-s + (137. − 66.3i)8-s + (310. + 95.7i)9-s + (3.88 + 51.8i)10-s + (−91.0 − 399. i)11-s + (67.3 − 171. i)12-s + (59.3 − 791. i)13-s + (717. − 108. i)14-s + (−144. − 133. i)15-s + (1.09e3 + 525. i)16-s + (−1.37e3 + 937. i)17-s + ⋯ |
| L(s) = 1 | + (0.694 + 0.871i)2-s + (−1.51 − 0.227i)3-s + (−0.0538 + 0.235i)4-s + (0.121 + 0.0830i)5-s + (−0.851 − 1.47i)6-s + (0.444 − 0.769i)7-s + (0.761 − 0.366i)8-s + (1.27 + 0.393i)9-s + (0.0122 + 0.163i)10-s + (−0.226 − 0.994i)11-s + (0.135 − 0.344i)12-s + (0.0973 − 1.29i)13-s + (0.978 − 0.147i)14-s + (−0.165 − 0.153i)15-s + (1.06 + 0.513i)16-s + (−1.15 + 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.22977 - 0.509833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22977 - 0.509833i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.96e3 + 9.13e3i)T \) |
| good | 2 | \( 1 + (-3.93 - 4.92i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (23.5 + 3.55i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (-6.81 - 4.64i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (-57.5 + 99.7i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (91.0 + 399. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-59.3 + 791. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (1.37e3 - 937. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (485. - 149. i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-1.97e3 + 1.82e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (3.78e3 - 570. i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (-1.51e3 + 3.86e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (-3.30e3 - 5.71e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-3.21e3 - 4.02e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (3.63e3 - 1.59e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-587. - 7.84e3i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-4.65e3 - 2.24e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.20e4 + 3.08e4i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (3.48e4 - 1.07e4i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (-4.42e4 - 4.10e4i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (1.43e3 - 1.91e4i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (-5.10e4 + 8.83e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.57e4 - 5.39e3i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (-6.10e3 - 920. i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (-3.48e4 - 1.52e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.93753822797425180512452024229, −13.56485831489164668274461503625, −12.74356422772088515108428390109, −10.98102652907908794042801037153, −10.57961437867149554430732971947, −7.929288874897202338941261110319, −6.50551585025133733364076485145, −5.72254965589507832789163766326, −4.45455784606848841689695754685, −0.72043478297303763588347432134,
1.96921993045692891432052438651, 4.42758472033489653375854564399, 5.32854446369100006512549610396, 7.04812509518266756228406578832, 9.331855363286541270510689376598, 10.95532232420539387528263548760, 11.54678422273360616595255089371, 12.36177420867198733990230483770, 13.48720239705165776517094140867, 15.10839584294443481603867045937
