| L(s) = 1 | + (−12.2 + 10.2i)2-s + (117. + 42.9i)3-s + (44.4 − 252. i)4-s + (21.9 + 124. i)5-s + (−1.88e3 + 686. i)6-s + (2.83e3 − 4.90e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−3.01e3 − 2.53e3i)9-s + (−1.55e3 − 1.30e3i)10-s + (−2.79e4 − 4.84e4i)11-s + (1.60e4 − 2.78e4i)12-s + (−6.35e4 + 2.31e4i)13-s + (1.57e4 + 8.93e4i)14-s + (−2.75e3 + 1.56e4i)15-s + (−6.15e4 − 2.24e4i)16-s + (−1.39e5 + 1.17e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.840 + 0.305i)3-s + (0.0868 − 0.492i)4-s + (0.0157 + 0.0891i)5-s + (−0.594 + 0.216i)6-s + (0.446 − 0.772i)7-s + (0.176 + 0.306i)8-s + (−0.153 − 0.128i)9-s + (−0.0490 − 0.0411i)10-s + (−0.576 − 0.998i)11-s + (0.223 − 0.387i)12-s + (−0.617 + 0.224i)13-s + (0.109 + 0.621i)14-s + (−0.0140 + 0.0797i)15-s + (−0.234 − 0.0855i)16-s + (−0.406 + 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.932024 - 0.763198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932024 - 0.763198i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (12.2 - 10.2i)T \) |
| 19 | \( 1 + (5.67e5 - 2.22e4i)T \) |
| good | 3 | \( 1 + (-117. - 42.9i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-21.9 - 124. i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-2.83e3 + 4.90e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.79e4 + 4.84e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (6.35e4 - 2.31e4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (1.39e5 - 1.17e5i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-2.84e5 + 1.61e6i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (1.42e6 + 1.19e6i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (-4.09e6 + 7.09e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 6.50e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-2.39e6 - 8.70e5i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-4.58e6 - 2.59e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (3.38e7 + 2.83e7i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-7.14e6 + 4.05e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (1.68e7 - 1.41e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (1.24e7 - 7.06e7i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (5.61e7 + 4.70e7i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (-2.94e7 - 1.66e8i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-8.67e7 - 3.15e7i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (3.89e8 + 1.41e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (2.31e7 - 4.00e7i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-5.30e8 + 1.92e8i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-6.04e8 + 5.07e8i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−14.39187624252157889400531257836, −13.20583163217391624212287222006, −11.21989398075506241239245129252, −10.13711783059336722515837840701, −8.742827666812170331254973139742, −7.924262152029223383514706804636, −6.33002868600983705353478163391, −4.38120693120450522509596607683, −2.56190198385848098518075998626, −0.44777495055388657134621473423,
1.83038662358306461177396498587, 2.83157955963985338205440348922, 4.99916124467905529437572237020, 7.26957941139458994553092978621, 8.382988332368570256038707592072, 9.359991303181652603660289563749, 10.81314965077238149913395016536, 12.20195122823594173592692013000, 13.19070196851652116114431140690, 14.60996961092120781949918985353
