| L(s) = 1 | + (−0.715 − 1.35i)2-s + (0.725 + 0.687i)3-s + (−0.188 + 0.278i)4-s + (−3.13 − 0.690i)5-s + (0.408 − 1.47i)6-s + (−2.11 − 1.60i)7-s + (−2.52 − 0.274i)8-s + (0.0541 + 0.998i)9-s + (1.31 + 4.73i)10-s + (2.12 − 5.34i)11-s + (−0.328 + 0.0723i)12-s + (0.173 − 3.20i)13-s + (−0.657 + 4.01i)14-s + (−1.80 − 2.65i)15-s + (1.68 + 4.23i)16-s + (−1.99 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.506 − 0.954i)2-s + (0.419 + 0.397i)3-s + (−0.0944 + 0.139i)4-s + (−1.40 − 0.308i)5-s + (0.166 − 0.601i)6-s + (−0.799 − 0.608i)7-s + (−0.893 − 0.0972i)8-s + (0.0180 + 0.332i)9-s + (0.415 + 1.49i)10-s + (0.642 − 1.61i)11-s + (−0.0948 + 0.0208i)12-s + (0.0482 − 0.889i)13-s + (−0.175 + 1.07i)14-s + (−0.465 − 0.686i)15-s + (0.421 + 1.05i)16-s + (−0.483 + 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133338 - 0.629002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133338 - 0.629002i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.725 - 0.687i)T \) |
| 59 | \( 1 + (-6.29 + 4.39i)T \) |
| good | 2 | \( 1 + (0.715 + 1.35i)T + (-1.12 + 1.65i)T^{2} \) |
| 5 | \( 1 + (3.13 + 0.690i)T + (4.53 + 2.09i)T^{2} \) |
| 7 | \( 1 + (2.11 + 1.60i)T + (1.87 + 6.74i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 5.34i)T + (-7.98 - 7.56i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 3.20i)T + (-12.9 - 1.40i)T^{2} \) |
| 17 | \( 1 + (1.99 - 1.51i)T + (4.54 - 16.3i)T^{2} \) |
| 19 | \( 1 + (-5.97 - 2.01i)T + (15.1 + 11.4i)T^{2} \) |
| 23 | \( 1 + (-0.758 - 0.456i)T + (10.7 + 20.3i)T^{2} \) |
| 29 | \( 1 + (1.90 - 3.59i)T + (-16.2 - 24.0i)T^{2} \) |
| 31 | \( 1 + (4.32 - 1.45i)T + (24.6 - 18.7i)T^{2} \) |
| 37 | \( 1 + (-2.35 + 0.255i)T + (36.1 - 7.95i)T^{2} \) |
| 41 | \( 1 + (-10.8 + 6.50i)T + (19.2 - 36.2i)T^{2} \) |
| 43 | \( 1 + (2.62 + 6.57i)T + (-31.2 + 29.5i)T^{2} \) |
| 47 | \( 1 + (-5.95 + 1.30i)T + (42.6 - 19.7i)T^{2} \) |
| 53 | \( 1 + (-0.313 + 1.12i)T + (-45.4 - 27.3i)T^{2} \) |
| 61 | \( 1 + (2.55 + 4.81i)T + (-34.2 + 50.4i)T^{2} \) |
| 67 | \( 1 + (10.5 + 1.14i)T + (65.4 + 14.4i)T^{2} \) |
| 71 | \( 1 + (15.0 - 3.30i)T + (64.4 - 29.8i)T^{2} \) |
| 73 | \( 1 + (-0.660 + 4.02i)T + (-69.1 - 23.3i)T^{2} \) |
| 79 | \( 1 + (-3.08 + 2.92i)T + (4.27 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-9.62 - 11.3i)T + (-13.4 + 81.9i)T^{2} \) |
| 89 | \( 1 + (1.38 - 2.60i)T + (-49.9 - 73.6i)T^{2} \) |
| 97 | \( 1 + (1.59 + 9.71i)T + (-91.9 + 30.9i)T^{2} \) |
| 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
−11.99349445706911224320652209626, −11.12022111912607658880814508592, −10.48099332196236188652916630659, −9.261053972846792861810151819973, −8.529515090066959284456470666617, −7.36910292058290768852517630446, −5.76954090553071741033594837033, −3.71408943082261446817169084325, −3.31028952227146623696952977024, −0.66591848071388121758037440168,
2.83018729563737489134716992672, 4.27280727277106770574091791072, 6.27310354616402986171874669915, 7.28414087772750656117543082015, 7.56962949682261034037579206890, 9.061612426101662852336189728702, 9.484150159109381318743959959494, 11.58879786262507160249503637061, 11.97208131565121365463915159332, 13.00207624492688935926170365481
