| L(s) = 1 | + (1.10 + 0.886i)2-s + (0.755 − 0.654i)3-s + (0.428 + 1.95i)4-s + (0.994 − 0.142i)5-s + (1.41 − 0.0516i)6-s + (1.86 − 4.07i)7-s + (−1.25 + 2.53i)8-s + (0.142 − 0.989i)9-s + (1.22 + 0.723i)10-s + (0.868 − 2.95i)11-s + (1.60 + 1.19i)12-s + (0.420 − 0.192i)13-s + (5.66 − 2.83i)14-s + (0.657 − 0.759i)15-s + (−3.63 + 1.67i)16-s + (−5.27 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (0.779 + 0.626i)2-s + (0.436 − 0.378i)3-s + (0.214 + 0.976i)4-s + (0.444 − 0.0639i)5-s + (0.576 − 0.0210i)6-s + (0.703 − 1.53i)7-s + (−0.445 + 0.895i)8-s + (0.0474 − 0.329i)9-s + (0.386 + 0.228i)10-s + (0.261 − 0.891i)11-s + (0.462 + 0.345i)12-s + (0.116 − 0.0533i)13-s + (1.51 − 0.758i)14-s + (0.169 − 0.196i)15-s + (−0.908 + 0.418i)16-s + (−1.27 − 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.79361 + 0.302057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79361 + 0.302057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.10 - 0.886i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (-0.616 - 4.75i)T \) |
| good | 5 | \( 1 + (-0.994 + 0.142i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.86 + 4.07i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.868 + 2.95i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.420 + 0.192i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (5.27 + 3.39i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.49 - 6.99i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.75 - 2.73i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.99 - 3.45i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-7.54 - 1.08i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0346 - 0.240i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.26 - 4.56i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 0.826T + 47T^{2} \) |
| 53 | \( 1 + (-3.02 - 1.37i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (5.22 - 2.38i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.24 - 4.54i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.44 + 4.93i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.91 + 1.14i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-10.4 + 6.73i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.04 - 2.29i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (17.5 + 2.52i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (9.73 + 11.2i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.54 + 10.7i)T + (-93.0 + 27.3i)T^{2} \) |
| 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
−11.14571396782517887098785411930, −9.862023438048240665457249807677, −8.794912771623103836602818995412, −7.78149788894023490007712870263, −7.32041117584331211413313951892, −6.27254355796441645323473972182, −5.27179501782832890218174517484, −4.10040043662577596350917852121, −3.26919027743879414894912313871, −1.53766090155045437061242405431,
2.04527414426817232546747976258, 2.54163308485129457315128959499, 4.15393266568647935129032050061, 4.97414792247523986232211204984, 5.85887887871849199857345404146, 6.89656032262762542062976834905, 8.394013813620768852821856466287, 9.259350012396881330030908836606, 9.772921949991147598430953531230, 11.05265397475877365463746343617
