| L(s) = 1 | + (−0.706 − 2.54i)2-s + (1.58 + 0.695i)3-s + (−4.25 + 2.56i)4-s + (3.25 − 0.176i)5-s + (0.649 − 4.52i)6-s + (−0.210 − 1.28i)7-s + (5.69 + 5.39i)8-s + (2.03 + 2.20i)9-s + (−2.74 − 8.15i)10-s + (−1.93 − 3.64i)11-s + (−8.53 + 1.10i)12-s + (−1.56 − 0.623i)13-s + (−3.11 + 1.43i)14-s + (5.28 + 1.98i)15-s + (5.03 − 9.50i)16-s + (3.37 + 0.553i)17-s + ⋯ |
| L(s) = 1 | + (−0.499 − 1.79i)2-s + (0.915 + 0.401i)3-s + (−2.12 + 1.28i)4-s + (1.45 − 0.0788i)5-s + (0.265 − 1.84i)6-s + (−0.0794 − 0.484i)7-s + (2.01 + 1.90i)8-s + (0.677 + 0.735i)9-s + (−0.868 − 2.57i)10-s + (−0.582 − 1.09i)11-s + (−2.46 + 0.317i)12-s + (−0.433 − 0.172i)13-s + (−0.831 + 0.384i)14-s + (1.36 + 0.512i)15-s + (1.25 − 2.37i)16-s + (0.819 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.750992 - 1.00755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.750992 - 1.00755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.58 - 0.695i)T \) |
| 59 | \( 1 + (-6.38 + 4.27i)T \) |
| good | 2 | \( 1 + (0.706 + 2.54i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-3.25 + 0.176i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.210 + 1.28i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (1.93 + 3.64i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (1.56 + 0.623i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (-3.37 - 0.553i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (5.39 - 6.35i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.326 + 0.248i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (6.81 + 1.89i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (0.0480 - 0.0408i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.21 - 3.39i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-2.06 - 2.71i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (4.01 + 2.12i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (-0.111 + 2.05i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (1.69 - 5.03i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (6.57 - 1.82i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (7.13 - 7.52i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-3.83 - 0.208i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (-2.50 - 5.41i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (0.654 + 0.964i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (9.57 - 2.10i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (2.24 - 8.09i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (1.16 - 2.51i)T + (-62.7 - 73.9i)T^{2} \) |
| 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
−12.53933973409890006379498007807, −11.00290055373525962167202224074, −10.14958573000616493586265957960, −9.865823236432923659470252657301, −8.753163089585978840595117838385, −7.930179361069343777161996600867, −5.60185119678316659814246137984, −4.01997125655933729770827606726, −2.85812630322532485462513935439, −1.71713942146382651062281924571,
2.18650409242823565445087617371, 4.77917920678784540548034559912, 5.89576108892864854404292905443, 6.89074730080337894016248826376, 7.67450412003461588424055094111, 8.956199120646181539300979695483, 9.428209356137202943460224217752, 10.25283165590769431739094155352, 12.73526857194660906715386148007, 13.29625935107075287054586309057
