L(s) = 1 | + (2.31 + 1.33i)2-s + (−0.417 − 0.723i)4-s + (−0.223 − 0.386i)5-s + (18.4 − 1.96i)7-s − 23.6i·8-s − 1.19i·10-s + (−34.2 − 19.7i)11-s + (68.4 − 39.5i)13-s + (45.3 + 20.0i)14-s + (28.3 − 49.0i)16-s − 9.74·17-s + 73.1i·19-s + (−0.186 + 0.323i)20-s + (−52.9 − 91.7i)22-s + (126. − 73.0i)23-s + ⋯ |
L(s) = 1 | + (0.819 + 0.473i)2-s + (−0.0522 − 0.0904i)4-s + (−0.0199 − 0.0345i)5-s + (0.994 − 0.106i)7-s − 1.04i·8-s − 0.0377i·10-s + (−0.939 − 0.542i)11-s + (1.46 − 0.843i)13-s + (0.865 + 0.383i)14-s + (0.442 − 0.766i)16-s − 0.139·17-s + 0.882i·19-s + (−0.00208 + 0.00361i)20-s + (−0.513 − 0.889i)22-s + (1.14 − 0.661i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.59893 - 0.628803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.59893 - 0.628803i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-18.4 + 1.96i)T \) |
good | 2 | \( 1 + (-2.31 - 1.33i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (0.223 + 0.386i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (34.2 + 19.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-68.4 + 39.5i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 9.74T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-126. + 73.0i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-134. - 77.4i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-9.87 + 5.70i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (53.3 + 92.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (45.6 - 78.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (276. - 479. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 239. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (126. + 218. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (342. + 197. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 23.4i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 923. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-280. + 485. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-281. + 487. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 644.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-427. - 246. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.35378777900953065640688150349, −10.86190082894990160480415506670, −10.43471941933265217170499744476, −8.730379362056367997552309828510, −7.957162648453665552584348350620, −6.52705022148399250293881812257, −5.51580588110466019825633672632, −4.65634130078318486003906494704, −3.26477376893487088790822363953, −1.03996844834528078227172848947,
1.79340590118785111655769803091, 3.25539914301255539237035353310, 4.59321306925687524777408415462, 5.33047123177782626548778550841, 6.98430543199736511779553841952, 8.242076844952793322464563493386, 9.005730396390981806735250298262, 10.71224187800166432757949521075, 11.31770599246509862032997343954, 12.15231409916994082346840867419