| L(s) = 1 | + (−2.62 + 3.87i)2-s + (−4.98 − 1.47i)3-s + (−5.15 − 12.9i)4-s + (1.79 − 3.87i)5-s + (18.8 − 15.4i)6-s + (−7.00 − 25.2i)7-s + (27.1 + 5.96i)8-s + (22.6 + 14.7i)9-s + (10.3 + 17.1i)10-s + (−11.3 − 10.7i)11-s + (6.56 + 72.0i)12-s + (−5.74 + 52.8i)13-s + (116. + 39.1i)14-s + (−14.6 + 16.6i)15-s + (−13.5 + 12.7i)16-s + (2.47 + 0.686i)17-s + ⋯ |
| L(s) = 1 | + (−0.929 + 1.37i)2-s + (−0.958 − 0.284i)3-s + (−0.644 − 1.61i)4-s + (0.160 − 0.346i)5-s + (1.28 − 1.04i)6-s + (−0.378 − 1.36i)7-s + (1.19 + 0.263i)8-s + (0.838 + 0.545i)9-s + (0.325 + 0.541i)10-s + (−0.311 − 0.294i)11-s + (0.157 + 1.73i)12-s + (−0.122 + 1.12i)13-s + (2.21 + 0.747i)14-s + (−0.252 + 0.286i)15-s + (−0.211 + 0.199i)16-s + (0.0352 + 0.00978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0327513 + 0.245292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0327513 + 0.245292i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.98 + 1.47i)T \) |
| 59 | \( 1 + (-307. - 332. i)T \) |
| good | 2 | \( 1 + (2.62 - 3.87i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (-1.79 + 3.87i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (7.00 + 25.2i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (11.3 + 10.7i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (5.74 - 52.8i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-2.47 - 0.686i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (47.3 + 35.9i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (6.59 + 12.4i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (41.0 - 27.8i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (42.1 + 55.4i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-36.7 - 166. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (303. + 160. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-147. - 155. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (266. - 123. i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (92.3 - 153. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (-550. - 372. i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (191. - 871. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-359. - 777. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (234. - 696. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (-29.2 + 539. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (56.9 - 347. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (331. + 489. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-493. - 1.46e3i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.91209969713163154694884238908, −11.44604993285368803930026938059, −10.42625035184457525458930791505, −9.602918696455764927464172468203, −8.419321053375620158663253365856, −7.15003062652679661077591283411, −6.79565439519029289074220448903, −5.59890709735755462007915149057, −4.38669934618740993194533793370, −1.09434948184183049956380689892,
0.20624788911009369652312254863, 2.11032336065956638542407889776, 3.38274655815400860366833979914, 5.21564815526758467891528241998, 6.35326298677534809812463243207, 8.049813343646716351769294737332, 9.172586411119748648221485116872, 10.05983262827901622505627710203, 10.66430949749487035358070370514, 11.63900588010500323034334384789
