| L(s) = 1 | + (0.319 − 2.22i)2-s + (0.415 + 0.909i)3-s + (−2.91 − 0.855i)4-s + (−1.82 − 2.10i)5-s + (2.15 − 0.632i)6-s + (2.85 + 1.83i)7-s + (−0.966 + 2.11i)8-s + (−0.654 + 0.755i)9-s + (−5.25 + 3.37i)10-s + (0.730 + 5.08i)11-s + (−0.432 − 3.00i)12-s + (−1.83 + 1.17i)13-s + (4.98 − 5.75i)14-s + (1.15 − 2.53i)15-s + (−0.717 − 0.461i)16-s + (2.48 − 0.729i)17-s + ⋯ |
| L(s) = 1 | + (0.225 − 1.57i)2-s + (0.239 + 0.525i)3-s + (−1.45 − 0.427i)4-s + (−0.815 − 0.940i)5-s + (0.879 − 0.258i)6-s + (1.07 + 0.693i)7-s + (−0.341 + 0.747i)8-s + (−0.218 + 0.251i)9-s + (−1.66 + 1.06i)10-s + (0.220 + 1.53i)11-s + (−0.124 − 0.867i)12-s + (−0.508 + 0.327i)13-s + (1.33 − 1.53i)14-s + (0.298 − 0.653i)15-s + (−0.179 − 0.115i)16-s + (0.602 − 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0933 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.666699 - 0.732157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.666699 - 0.732157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (3.18 + 3.58i)T \) |
| good | 2 | \( 1 + (-0.319 + 2.22i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (1.82 + 2.10i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-2.85 - 1.83i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.730 - 5.08i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.83 - 1.17i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 0.729i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (2.21 + 0.651i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.12 - 1.50i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.71 + 8.14i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.43 - 1.65i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.199 + 0.230i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.81 - 8.35i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 4.99T + 47T^{2} \) |
| 53 | \( 1 + (-6.89 - 4.43i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 2.58i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.43 + 5.32i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.130 + 0.906i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.189 + 1.31i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (9.77 + 2.86i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.9 + 7.70i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (1.75 - 2.03i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.324 + 0.709i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.43 - 2.80i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.45067988934701348448806385516, −12.84300676242044580275289664689, −12.06138175390902576149554096780, −11.45729342889037642579160109746, −10.04148090033379861654888926641, −9.087197026818617720178628647371, −7.86713253398324142587264498559, −4.85057124130277007294785231597, −4.24049378106564740804285567044, −2.11059027743469470837226898081,
3.75935971435840361567807914181, 5.58224649750323726850935943673, 6.95449845366076246164790959619, 7.76860915033260545655944799291, 8.484694102770456717667977819851, 10.71407476117809867381029228571, 11.74072080819603617070519647855, 13.51148506666838372940572480201, 14.31563296010832075814311736864, 14.82858695691295216045295762575
