| L(s) = 1 | + (−6.12 + 5.14i)2-s + (10.5 + 3.82i)3-s + (11.1 − 63.0i)4-s + (7.39 + 41.9i)5-s + (−84.0 + 30.6i)6-s + (58.1 − 100. i)7-s + (256. + 443. i)8-s + (−1.57e3 − 1.32e3i)9-s + (−260. − 218. i)10-s + (−915. − 1.58e3i)11-s + (357. − 620. i)12-s + (2.42e3 − 882. i)13-s + (161. + 915. i)14-s + (−82.7 + 469. i)15-s + (−3.84e3 − 1.40e3i)16-s + (2.31e4 − 1.94e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.224 + 0.0818i)3-s + (0.0868 − 0.492i)4-s + (0.0264 + 0.150i)5-s + (−0.158 + 0.0578i)6-s + (0.0640 − 0.110i)7-s + (0.176 + 0.306i)8-s + (−0.722 − 0.606i)9-s + (−0.0825 − 0.0692i)10-s + (−0.207 − 0.359i)11-s + (0.0598 − 0.103i)12-s + (0.306 − 0.111i)13-s + (0.0157 + 0.0891i)14-s + (−0.00632 + 0.0358i)15-s + (−0.234 − 0.0855i)16-s + (1.14 − 0.960i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.10251 - 0.476293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10251 - 0.476293i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.12 - 5.14i)T \) |
| 19 | \( 1 + (-2.47e4 + 1.67e4i)T \) |
| good | 3 | \( 1 + (-10.5 - 3.82i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-7.39 - 41.9i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-58.1 + 100. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (915. + 1.58e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.42e3 + 882. i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.31e4 + 1.94e4i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.09e4 + 6.23e4i)T + (-3.19e9 - 1.16e9i)T^{2} \) |
| 29 | \( 1 + (3.32e4 + 2.79e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (4.31e4 - 7.46e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.94e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.54e5 + 9.26e4i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (2.04e4 + 1.16e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (3.11e5 + 2.61e5i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-8.21e4 + 4.65e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (5.73e5 - 4.81e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-3.04e5 + 1.72e6i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-1.09e6 - 9.17e5i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (2.64e5 + 1.50e6i)T + (-8.54e12 + 3.11e12i)T^{2} \) |
| 73 | \( 1 + (1.06e6 + 3.89e5i)T + (8.46e12 + 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-4.17e6 - 1.51e6i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (8.35e5 - 1.44e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (4.08e6 - 1.48e6i)T + (3.38e13 - 2.84e13i)T^{2} \) |
| 97 | \( 1 + (4.12e6 - 3.45e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−14.65928821287145917189455365136, −13.82533605671622566946709420590, −12.06406460826981151665067481810, −10.75637896276416718916202441082, −9.402808676288430071152113085679, −8.297263073262632261119212791559, −6.85730362708859449584566736098, −5.34824304647006536278906148349, −3.06010242212633803636204681459, −0.65313194135711193344998368736,
1.55742430222114422150674139535, 3.32432495673560921676964962885, 5.47270067184068381310692784250, 7.53571745392782934511551357204, 8.620518558967706316450094476924, 9.963635892785974686389600661739, 11.19652779593677702021093520440, 12.39493539332089395981390051953, 13.63663510200311698960545508761, 14.89385138190342108084389885014
