| L(s) = 1 | + (−0.542 + 1.14i)2-s + (2.23 + 2.05i)3-s + (0.252 + 0.309i)4-s + (0.809 + 2.08i)5-s + (−3.56 + 1.43i)6-s + (−4.71 − 1.36i)7-s + (−2.94 + 0.726i)8-s + (0.499 + 6.18i)9-s + (−2.82 − 0.205i)10-s + (−1.52 − 1.79i)11-s + (−0.0731 + 1.20i)12-s + (3.53 + 0.726i)13-s + (4.12 − 4.65i)14-s + (−2.48 + 6.31i)15-s + (0.608 − 2.98i)16-s + (1.93 + 3.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.383 + 0.808i)2-s + (1.28 + 1.18i)3-s + (0.126 + 0.154i)4-s + (0.362 + 0.932i)5-s + (−1.45 + 0.585i)6-s + (−1.78 − 0.516i)7-s + (−1.04 + 0.256i)8-s + (0.166 + 2.06i)9-s + (−0.892 − 0.0648i)10-s + (−0.459 − 0.539i)11-s + (−0.0211 + 0.349i)12-s + (0.979 + 0.201i)13-s + (1.10 − 1.24i)14-s + (−0.641 + 1.63i)15-s + (0.152 − 0.745i)16-s + (0.468 + 0.851i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.372699 - 1.63944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372699 - 1.63944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.809 - 2.08i)T \) |
| 13 | \( 1 + (-3.53 - 0.726i)T \) |
| good | 2 | \( 1 + (0.542 - 1.14i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-2.23 - 2.05i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (4.71 + 1.36i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (1.52 + 1.79i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.93 - 3.51i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-3.97 + 1.06i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.35 + 5.04i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.66 + 1.26i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-5.29 - 4.15i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-0.869 - 1.15i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (4.44 + 4.10i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.461 - 3.25i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (-7.08 + 2.68i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (6.94 - 11.4i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (2.18 + 1.44i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-0.0997 + 0.103i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (5.75 - 7.05i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (-1.12 + 4.99i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (3.48 - 5.05i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-5.71 + 2.16i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-4.63 - 8.83i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (7.02 + 1.88i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (12.6 - 4.21i)T + (77.5 - 58.2i)T^{2} \) |
| 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
−10.37261273086970343698472989045, −9.676447730026243873834454302938, −8.994127671966562305549615879386, −8.247182990052051947271694288441, −7.32887134636645040538233806847, −6.48704789339441114345409724866, −5.75637340428907283884676454851, −3.95898706504656088304441148089, −3.16446744008933396875265731900, −2.82434234846222823878876912041,
0.77741827846034177393476285470, 1.86925141590126408732083720173, 2.90555466048149479634822346804, 3.46500929003272266761256805309, 5.57522188117191114112729779940, 6.31301707991897054432945071963, 7.25654180315443120629913758313, 8.227516675237517592369748228863, 9.267701113154722153573095168418, 9.398878625555803252682816173336
