| 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
−13.00207624492688935926170365481, −11.97208131565121365463915159332, −11.58879786262507160249503637061, −9.484150159109381318743959959494, −9.061612426101662852336189728702, −7.56962949682261034037579206890, −7.28414087772750656117543082015, −6.27310354616402986171874669915, −4.27280727277106770574091791072, −2.83018729563737489134716992672,
0.66591848071388121758037440168, 3.31028952227146623696952977024, 3.71408943082261446817169084325, 5.76954090553071741033594837033, 7.36910292058290768852517630446, 8.529515090066959284456470666617, 9.261053972846792861810151819973, 10.48099332196236188652916630659, 11.12022111912607658880814508592, 11.99349445706911224320652209626
