| L(s) = 1 | + (0.460 − 0.679i)2-s + (−1.72 − 0.0951i)3-s + (0.490 + 1.23i)4-s + (−0.472 + 1.02i)5-s + (−0.861 + 1.13i)6-s + (0.500 + 1.80i)7-s + (2.66 + 0.587i)8-s + (2.98 + 0.328i)9-s + (0.475 + 0.791i)10-s + (1.67 + 1.58i)11-s + (−0.731 − 2.17i)12-s + (−0.00701 + 0.0645i)13-s + (1.45 + 0.490i)14-s + (0.913 − 1.72i)15-s + (−0.297 + 0.281i)16-s + (−1.12 − 0.311i)17-s + ⋯ |
| L(s) = 1 | + (0.325 − 0.480i)2-s + (−0.998 − 0.0549i)3-s + (0.245 + 0.615i)4-s + (−0.211 + 0.456i)5-s + (−0.351 + 0.461i)6-s + (0.189 + 0.681i)7-s + (0.942 + 0.207i)8-s + (0.993 + 0.109i)9-s + (0.150 + 0.250i)10-s + (0.503 + 0.477i)11-s + (−0.211 − 0.628i)12-s + (−0.00194 + 0.0178i)13-s + (0.389 + 0.131i)14-s + (0.235 − 0.444i)15-s + (−0.0743 + 0.0704i)16-s + (−0.272 − 0.0755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05128 + 0.322962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05128 + 0.322962i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.0951i)T \) |
| 59 | \( 1 + (-7.53 - 1.49i)T \) |
| good | 2 | \( 1 + (-0.460 + 0.679i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (0.472 - 1.02i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.500 - 1.80i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-1.67 - 1.58i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.00701 - 0.0645i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (1.12 + 0.311i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (2.84 + 2.16i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.0691 + 0.130i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-4.18 + 2.83i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (1.05 + 1.38i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (1.76 + 8.01i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (-3.40 - 1.80i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (7.42 + 7.83i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (5.61 - 2.59i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-4.90 + 8.16i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-5.16 - 3.49i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-0.0126 + 0.0576i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (0.791 + 1.71i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-3.52 + 10.4i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (0.795 - 14.6i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (-1.95 + 11.9i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (9.17 + 13.5i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-4.38 - 12.9i)T + (-77.2 + 58.7i)T^{2} \) |
| 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
−12.53784502840843623062931438463, −11.74144066511329541634311417735, −11.18369833527530003791173958845, −10.19694290951506587848512590404, −8.738759660016756120841184130411, −7.35812187672877078152987799075, −6.54195126891709513971524681568, −5.07072446082841323772659098015, −3.90087004714297475796861249129, −2.19828098043571355092894589505,
1.18465539678530899731673503558, 4.18348399046105882587947709980, 5.06961654448859368002105306340, 6.26287513386453059206388563636, 6.97477715433849254498267597975, 8.359512849415554723611797412019, 9.924210979969269110873419148377, 10.69283670994144495725694722854, 11.52695004623396180408687353125, 12.59834148430243746272286028816
