| L(s) = 1 | + (−5.19 + 3.00i)2-s + (−1.61 + 0.934i)3-s + (10.0 − 17.3i)4-s + (−20.3 − 35.1i)5-s + (5.61 − 9.71i)6-s − 11.8·7-s + 24.0i·8-s + (−38.7 + 67.1i)9-s + (211. + 121. i)10-s − 91.6·11-s + 37.4i·12-s + (−81.2 − 46.8i)13-s + (61.6 − 35.5i)14-s + (65.7 + 37.9i)15-s + (87.8 + 152. i)16-s + (−205. − 356. i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 + 0.750i)2-s + (−0.179 + 0.103i)3-s + (0.625 − 1.08i)4-s + (−0.812 − 1.40i)5-s + (0.155 − 0.269i)6-s − 0.242·7-s + 0.376i·8-s + (−0.478 + 0.828i)9-s + (2.11 + 1.21i)10-s − 0.757·11-s + 0.259i·12-s + (−0.480 − 0.277i)13-s + (0.314 − 0.181i)14-s + (0.292 + 0.168i)15-s + (0.343 + 0.594i)16-s + (−0.711 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0830246 - 0.139003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0830246 - 0.139003i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-333. - 139. i)T \) |
| good | 2 | \( 1 + (5.19 - 3.00i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (1.61 - 0.934i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (20.3 + 35.1i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + 11.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 91.6T + 1.46e4T^{2} \) |
| 13 | \( 1 + (81.2 + 46.8i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (205. + 356. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (168. - 291. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-932. - 538. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + 1.78e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 240. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.87e3 - 1.08e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.14e3 + 1.97e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-818. + 1.41e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-108. - 62.8i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.42e3 + 1.97e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.09e3 - 1.89e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.58e3 - 915. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-994. + 574. i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (2.47e3 + 4.28e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-103. + 60.0i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.03e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (661. + 381. i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-3.75e3 + 2.16e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−17.07274630324654673341624229846, −16.22785535543157572846909267550, −15.60468000777892511168797297223, −13.30991044939972886896053486393, −11.69217616119373879112589849814, −9.876986267810012400857926595958, −8.512942883739957345312812283002, −7.56626140776977057432150252377, −5.15063494567016896416000281776, −0.21847194291957140268237958099,
2.98389855025897535345658538559, 6.83304536315025630997585177445, 8.362855958937209497007742058166, 10.09497264695077601629590084643, 11.09542762869046920251057648958, 12.14875737902339890519225755935, 14.46582893417409744237921043078, 15.76303639405729301277181898961, 17.53208787730263518250061296214, 18.24622682654572471878384848577
