| 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
−13.87237922368551909424842310041, −12.60101199517703002851985174110, −11.28430206950584321522745917414, −10.46464066690459580535159161244, −9.269729755984045324135075041004, −8.765200819701386704347273449330, −7.38022714183110697242009781081, −5.80112781434154559381475433484, −3.95619283489636826910107381957, −2.51589671024885738769020426549,
1.59069469670810495414271526190, 3.65279498717152511685221818015, 5.97938660373605057164414104317, 6.97966986224728748397334635843, 8.128790808843177828125050761352, 8.962823896118600720815124304168, 10.11160199315165830035854240482, 11.31556029914156330776881386345, 12.69330344706213090247376168137, 13.36196003728156224126858140418
