| L(s) = 1 | + (7.51 + 2.73i)2-s + (8.60 + 48.7i)3-s + (49.0 + 41.1i)4-s + (379. − 318. i)5-s + (−68.8 + 390. i)6-s + (612. − 1.06e3i)7-s + (256. + 443. i)8-s + (−250. + 91.0i)9-s + (3.72e3 − 1.35e3i)10-s + (35.5 + 61.5i)11-s + (−1.58e3 + 2.74e3i)12-s + (−2.40e3 + 1.36e4i)13-s + (7.50e3 − 6.29e3i)14-s + (1.87e4 + 1.57e4i)15-s + (711. + 4.03e3i)16-s + (−1.06e4 − 3.89e3i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.183 + 1.04i)3-s + (0.383 + 0.321i)4-s + (1.35 − 1.13i)5-s + (−0.130 + 0.737i)6-s + (0.674 − 1.16i)7-s + (0.176 + 0.306i)8-s + (−0.114 + 0.0416i)9-s + (1.17 − 0.428i)10-s + (0.00805 + 0.0139i)11-s + (−0.264 + 0.458i)12-s + (−0.304 + 1.72i)13-s + (0.730 − 0.613i)14-s + (1.43 + 1.20i)15-s + (0.0434 + 0.246i)16-s + (−0.528 − 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.39068 + 0.964198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.39068 + 0.964198i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 - 2.73i)T \) |
| 19 | \( 1 + (2.41e4 + 1.75e4i)T \) |
| good | 3 | \( 1 + (-8.60 - 48.7i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (-379. + 318. i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-612. + 1.06e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-35.5 - 61.5i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (2.40e3 - 1.36e4i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.06e4 + 3.89e3i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-685. - 575. i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (2.22e4 - 8.08e3i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.04e5 - 1.81e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (6.81e4 + 3.86e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (4.08e5 - 3.42e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (6.73e5 - 2.45e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (1.11e6 + 9.37e5i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (3.06e5 + 1.11e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-2.14e6 - 1.80e6i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (1.35e5 - 4.94e4i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-2.14e6 + 1.79e6i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (2.77e5 + 1.57e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (5.15e5 + 2.92e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (4.20e6 - 7.27e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.38e6 + 7.86e6i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (5.13e6 + 1.86e6i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−14.62902504108389315018343015529, −13.85067857760019486466908736534, −12.86975649029728440372247203737, −11.12704466128102755632761042463, −9.815153638128547166889226574336, −8.822200316281289487424781323931, −6.74759305143295526412367423734, −4.89656417626145525260091137045, −4.30412555391383228006378756091, −1.70686094751140060039636488040,
1.86075292939356649733644643539, 2.66940545408828749982108408535, 5.52343045712841719367440106705, 6.45116700989221894079874038319, 7.996368911060675673109082162367, 9.942660492595684846149750511971, 11.14464654668652485104629147394, 12.66026754586658782155931627117, 13.32327973190037162659675846974, 14.59770122045496590097825711196
