| L(s) = 1 | + (−0.967 + 2.65i)2-s + (−10.7 + 1.88i)3-s + (−6.12 − 5.14i)4-s + (8.30 − 6.96i)5-s + (5.33 − 30.2i)6-s + (31.7 − 54.9i)7-s + (19.5 − 11.3i)8-s + (34.8 − 12.6i)9-s + (10.4 + 28.7i)10-s + (−76.3 − 132. i)11-s + (75.2 + 43.4i)12-s + (25.4 + 4.48i)13-s + (115. + 137. i)14-s + (−75.6 + 90.1i)15-s + (11.1 + 63.0i)16-s + (−328. − 119. i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−1.18 + 0.209i)3-s + (−0.383 − 0.321i)4-s + (0.332 − 0.278i)5-s + (0.148 − 0.840i)6-s + (0.647 − 1.12i)7-s + (0.306 − 0.176i)8-s + (0.430 − 0.156i)9-s + (0.104 + 0.287i)10-s + (−0.630 − 1.09i)11-s + (0.522 + 0.301i)12-s + (0.150 + 0.0265i)13-s + (0.588 + 0.701i)14-s + (−0.336 + 0.400i)15-s + (0.0434 + 0.246i)16-s + (−1.13 − 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.505931 - 0.348511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.505931 - 0.348511i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.967 - 2.65i)T \) |
| 19 | \( 1 + (-14.0 + 360. i)T \) |
| good | 3 | \( 1 + (10.7 - 1.88i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-8.30 + 6.96i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (-31.7 + 54.9i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (76.3 + 132. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-25.4 - 4.48i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (328. + 119. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (114. + 96.3i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-186. - 512. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (373. + 215. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.75e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-758. + 133. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-2.06e3 + 1.73e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (3.02e3 - 1.09e3i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (1.11e3 - 1.32e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (1.93e3 - 5.32e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-4.26e3 - 3.57e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (2.06e3 + 5.68e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (-3.24e3 - 3.86e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (599. + 3.39e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (5.64e3 - 995. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-4.53e3 + 7.86e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-7.87e3 - 1.38e3i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-1.78e3 + 4.90e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−15.78612084405178328753323071054, −14.09969785819454363258092176536, −13.17266278729133671476186558657, −11.21662678444134227212420262088, −10.67150220944700768288386133393, −8.901012250759561248232271881481, −7.29699004713252597382064813247, −5.82984083924333862913132920999, −4.65834620042769957857995128975, −0.51964652220841534418433075583,
2.07668265412097832721076675030, 4.90602671484577481747846461466, 6.27099111518461590319237879106, 8.226288005284009001325742206500, 9.883101658964999099329842370057, 11.07074957706119069302365150312, 11.99263289081465532746427280147, 12.88119979660092777049494582563, 14.61082563288924348879451732181, 15.90811618722738819234727782380
