| L(s) = 1 | + (−0.939 + 0.342i)2-s + (1.08 − 1.34i)3-s + (0.766 − 0.642i)4-s + (0.343 − 0.408i)5-s + (−0.560 + 1.63i)6-s + (−0.716 − 1.24i)7-s + (−0.500 + 0.866i)8-s + (−0.637 − 2.93i)9-s + (−0.182 + 0.501i)10-s + (1.25 + 0.725i)11-s + (−0.0343 − 1.73i)12-s + (2.94 − 0.519i)13-s + (1.09 + 0.921i)14-s + (−0.178 − 0.907i)15-s + (0.173 − 0.984i)16-s + (1.89 + 5.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.627 − 0.778i)3-s + (0.383 − 0.321i)4-s + (0.153 − 0.182i)5-s + (−0.228 + 0.669i)6-s + (−0.270 − 0.469i)7-s + (−0.176 + 0.306i)8-s + (−0.212 − 0.977i)9-s + (−0.0577 + 0.158i)10-s + (0.378 + 0.218i)11-s + (−0.00990 − 0.499i)12-s + (0.817 − 0.144i)13-s + (0.293 + 0.246i)14-s + (−0.0461 − 0.234i)15-s + (0.0434 − 0.246i)16-s + (0.459 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.888601 - 0.324094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.888601 - 0.324094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-1.08 + 1.34i)T \) |
| 19 | \( 1 + (4.35 - 0.143i)T \) |
| good | 5 | \( 1 + (-0.343 + 0.408i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.716 + 1.24i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.25 - 0.725i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 + 0.519i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 5.20i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.396 - 0.472i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (4.97 + 1.81i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.28 - 2.47i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.41iT - 37T^{2} \) |
| 41 | \( 1 + (1.37 - 7.78i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.88 - 4.10i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.37 + 12.0i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.41 - 1.18i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (1.75 - 0.639i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-9.02 + 7.57i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.17 + 8.71i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (9.59 + 8.05i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.80 - 15.9i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (7.87 + 1.38i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.29 + 2.48i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.832 + 4.71i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.83 + 7.79i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−13.36196003728156224126858140418, −12.69330344706213090247376168137, −11.31556029914156330776881386345, −10.11160199315165830035854240482, −8.962823896118600720815124304168, −8.128790808843177828125050761352, −6.97966986224728748397334635843, −5.97938660373605057164414104317, −3.65279498717152511685221818015, −1.59069469670810495414271526190,
2.51589671024885738769020426549, 3.95619283489636826910107381957, 5.80112781434154559381475433484, 7.38022714183110697242009781081, 8.765200819701386704347273449330, 9.269729755984045324135075041004, 10.46464066690459580535159161244, 11.28430206950584321522745917414, 12.60101199517703002851985174110, 13.87237922368551909424842310041
