| L(s) = 1 | + (−0.334 + 0.705i)2-s + (−2.30 − 2.12i)3-s + (0.879 + 1.07i)4-s + (−2.22 − 0.233i)5-s + (2.27 − 0.914i)6-s + (−0.816 − 0.236i)7-s + (−2.57 + 0.633i)8-s + (0.549 + 6.80i)9-s + (0.909 − 1.49i)10-s + (−0.783 − 0.921i)11-s + (0.263 − 4.35i)12-s + (0.957 + 3.47i)13-s + (0.440 − 0.496i)14-s + (4.62 + 5.26i)15-s + (−0.142 + 0.697i)16-s + (−0.560 − 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.236 + 0.498i)2-s + (−1.33 − 1.22i)3-s + (0.439 + 0.538i)4-s + (−0.994 − 0.104i)5-s + (0.927 − 0.373i)6-s + (−0.308 − 0.0893i)7-s + (−0.908 + 0.224i)8-s + (0.183 + 2.26i)9-s + (0.287 − 0.471i)10-s + (−0.236 − 0.277i)11-s + (0.0759 − 1.25i)12-s + (0.265 + 0.964i)13-s + (0.117 − 0.132i)14-s + (1.19 + 1.35i)15-s + (−0.0356 + 0.174i)16-s + (−0.135 − 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.512955 - 0.174427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.512955 - 0.174427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.22 + 0.233i)T \) |
| 13 | \( 1 + (-0.957 - 3.47i)T \) |
| good | 2 | \( 1 + (0.334 - 0.705i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (2.30 + 2.12i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (0.816 + 0.236i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (0.783 + 0.921i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (0.560 + 1.01i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.675 + 0.180i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.920 + 3.43i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.98 + 2.36i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-6.20 - 4.85i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (2.44 + 3.25i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (0.675 + 0.623i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.01 + 7.18i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (-8.44 + 3.20i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-6.20 + 10.2i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 7.44i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-3.01 + 3.14i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (3.27 - 4.01i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (-2.06 + 9.17i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (3.45 - 4.99i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-10.0 + 3.81i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-2.22 - 4.23i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-3.67 - 0.985i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-8.99 + 3.00i)T + (77.5 - 58.2i)T^{2} \) |
| 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
−10.44341438839760909188962382964, −8.817730706918732715383974250211, −8.170108366109842776877678518955, −7.11359644467613961181481676690, −6.98263440101577535339761849356, −6.07005392710494238327384877186, −5.05331302341158204823933044094, −3.73562623078191949152694363720, −2.23654114365951850181520054534, −0.50120667695451861093089064211,
0.816307518691229144150517467563, 3.00528421371673427284737352620, 3.91884680865049552965294316469, 4.99482456221826486836076287543, 5.78642669978966384159456905997, 6.55706203835046913827297559998, 7.68022650092852873716101029749, 8.994821691285327601634740355453, 9.836394292278920904928814928065, 10.41980314879480309445008675212
