| L(s) = 1 | + (3.44 + 2.74i)2-s + (−1.73 − 11.5i)3-s + (0.765 + 3.35i)4-s + (−7.49 − 10.9i)5-s + (25.6 − 44.4i)6-s + (25.5 − 14.7i)7-s + (24.0 − 49.8i)8-s + (−51.8 + 16.0i)9-s + (4.38 − 58.5i)10-s + (−15.9 + 69.8i)11-s + (37.2 − 14.6i)12-s + (2.71 + 36.1i)13-s + (128. + 19.4i)14-s + (−113. + 105. i)15-s + (269. − 129. i)16-s + (257. + 175. i)17-s + ⋯ |
| L(s) = 1 | + (0.861 + 0.687i)2-s + (−0.192 − 1.27i)3-s + (0.0478 + 0.209i)4-s + (−0.299 − 0.439i)5-s + (0.712 − 1.23i)6-s + (0.522 − 0.301i)7-s + (0.375 − 0.779i)8-s + (−0.640 + 0.197i)9-s + (0.0438 − 0.585i)10-s + (−0.131 + 0.576i)11-s + (0.258 − 0.101i)12-s + (0.0160 + 0.214i)13-s + (0.657 + 0.0990i)14-s + (−0.504 + 0.467i)15-s + (1.05 − 0.507i)16-s + (0.889 + 0.606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.85912 - 0.960157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85912 - 0.960157i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.03e3 + 1.53e3i)T \) |
| good | 2 | \( 1 + (-3.44 - 2.74i)T + (3.56 + 15.5i)T^{2} \) |
| 3 | \( 1 + (1.73 + 11.5i)T + (-77.4 + 23.8i)T^{2} \) |
| 5 | \( 1 + (7.49 + 10.9i)T + (-228. + 581. i)T^{2} \) |
| 7 | \( 1 + (-25.5 + 14.7i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (15.9 - 69.8i)T + (-1.31e4 - 6.35e3i)T^{2} \) |
| 13 | \( 1 + (-2.71 - 36.1i)T + (-2.82e4 + 4.25e3i)T^{2} \) |
| 17 | \( 1 + (-257. - 175. i)T + (3.05e4 + 7.77e4i)T^{2} \) |
| 19 | \( 1 + (104. - 340. i)T + (-1.07e5 - 7.34e4i)T^{2} \) |
| 23 | \( 1 + (172. + 159. i)T + (2.09e4 + 2.79e5i)T^{2} \) |
| 29 | \( 1 + (65.7 - 436. i)T + (-6.75e5 - 2.08e5i)T^{2} \) |
| 31 | \( 1 + (-382. - 974. i)T + (-6.76e5 + 6.28e5i)T^{2} \) |
| 37 | \( 1 + (-1.34e3 - 774. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-617. + 774. i)T + (-6.28e5 - 2.75e6i)T^{2} \) |
| 47 | \( 1 + (332. + 1.45e3i)T + (-4.39e6 + 2.11e6i)T^{2} \) |
| 53 | \( 1 + (121. - 1.61e3i)T + (-7.80e6 - 1.17e6i)T^{2} \) |
| 59 | \( 1 + (5.27e3 - 2.53e3i)T + (7.55e6 - 9.47e6i)T^{2} \) |
| 61 | \( 1 + (-3.32e3 - 1.30e3i)T + (1.01e7 + 9.41e6i)T^{2} \) |
| 67 | \( 1 + (-5.06e3 - 1.56e3i)T + (1.66e7 + 1.13e7i)T^{2} \) |
| 71 | \( 1 + (-8.01 - 8.64i)T + (-1.89e6 + 2.53e7i)T^{2} \) |
| 73 | \( 1 + (6.45e3 - 483. i)T + (2.80e7 - 4.23e6i)T^{2} \) |
| 79 | \( 1 + (4.61e3 + 8.00e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (7.87e3 - 1.18e3i)T + (4.53e7 - 1.39e7i)T^{2} \) |
| 89 | \( 1 + (-618. - 4.10e3i)T + (-5.99e7 + 1.84e7i)T^{2} \) |
| 97 | \( 1 + (-2.82e3 + 1.23e4i)T + (-7.97e7 - 3.84e7i)T^{2} \) |
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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
−14.71568167299311467212631431018, −13.92269035329614567962630130710, −12.70463891221878351787392712328, −12.17203555532778799078494927975, −10.28521984759628544187144828217, −8.128560579188921905996585055389, −7.11722413875326822108056507663, −5.88978315564315941748595190452, −4.36408954417940228146743855896, −1.29669268729810663305598910434,
2.98615628386379396342559719430, 4.30563151405174259699786304711, 5.46911343037629599019487846012, 7.932379766882571162675115155682, 9.601669155023897014586590389215, 11.05117789518632983193263714934, 11.45369749386459432481784336268, 12.99856893250985865068732840190, 14.30748118085965427131515786933, 15.17712111806623454982727147982
