| L(s) = 1 | + (3.75 + 1.36i)2-s + (4.62 + 26.2i)3-s + (12.2 + 10.2i)4-s + (14.7 − 12.3i)5-s + (−18.5 + 105. i)6-s + (4.52 − 7.84i)7-s + (32.0 + 55.4i)8-s + (−439. + 159. i)9-s + (72.3 − 26.3i)10-s + (−79.1 − 137. i)11-s + (−213. + 369. i)12-s + (−12.8 + 72.5i)13-s + (27.7 − 23.2i)14-s + (392. + 329. i)15-s + (44.4 + 252. i)16-s + (1.77e3 + 645. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.296 + 1.68i)3-s + (0.383 + 0.321i)4-s + (0.263 − 0.221i)5-s + (−0.209 + 1.19i)6-s + (0.0349 − 0.0605i)7-s + (0.176 + 0.306i)8-s + (−1.80 + 0.657i)9-s + (0.228 − 0.0832i)10-s + (−0.197 − 0.341i)11-s + (−0.427 + 0.740i)12-s + (−0.0210 + 0.119i)13-s + (0.0378 − 0.0317i)14-s + (0.450 + 0.378i)15-s + (0.0434 + 0.246i)16-s + (1.48 + 0.541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.48159 + 2.02560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48159 + 2.02560i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.75 - 1.36i)T \) |
| 19 | \( 1 + (-1.20e3 + 1.00e3i)T \) |
| good | 3 | \( 1 + (-4.62 - 26.2i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (-14.7 + 12.3i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-4.52 + 7.84i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (79.1 + 137. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (12.8 - 72.5i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-1.77e3 - 645. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 23 | \( 1 + (2.36e3 + 1.98e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-6.28e3 + 2.28e3i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (-3.47e3 + 6.01e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.05e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.28e3 - 7.29e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-7.51e3 + 6.30e3i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (7.97e3 - 2.90e3i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 + (2.06e4 + 1.73e4i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (1.44e4 + 5.24e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-5.94e3 - 4.99e3i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (3.27e4 - 1.19e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-8.82e3 + 7.40e3i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + (1.45e4 + 8.27e4i)T + (-1.94e9 + 7.09e8i)T^{2} \) |
| 79 | \( 1 + (1.41e4 + 8.00e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (3.27e4 - 5.67e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.38e3 + 1.91e4i)T + (-5.24e9 - 1.90e9i)T^{2} \) |
| 97 | \( 1 + (1.04e5 + 3.78e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−15.62802213112267844340180119105, −14.56876343541733507785591906142, −13.68820977607316129870256696092, −11.97166933654561335480482954400, −10.58523287996480518364781570552, −9.559984992081729241349433622398, −8.117888982689309583887335193112, −5.78699268665822945740189118506, −4.56583928331385505610241969902, −3.18462570519645571246906311182,
1.37426363035482047066224496840, 2.94914509073006604922278464420, 5.62265455572053469713274969928, 6.96040556326451524384846366892, 8.078781737184465714803473405712, 10.09920466093510733699209614674, 11.99768554628797928872281680181, 12.40548205679675521926498709870, 13.98241634372922804697287552077, 14.10944121196498232906028664780
