| L(s) = 1 | + (0.173 − 0.984i)2-s + (0.338 + 0.284i)3-s + (−0.939 − 0.342i)4-s + (0.837 − 0.304i)5-s + (0.338 − 0.284i)6-s + (−1.26 − 2.18i)7-s + (−0.5 + 0.866i)8-s + (−0.486 − 2.76i)9-s + (−0.154 − 0.877i)10-s + (−0.975 + 1.69i)11-s + (−0.221 − 0.383i)12-s + (4.94 − 4.15i)13-s + (−2.36 + 0.861i)14-s + (0.370 + 0.134i)15-s + (0.766 + 0.642i)16-s + (−0.594 + 3.37i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.195 + 0.164i)3-s + (−0.469 − 0.171i)4-s + (0.374 − 0.136i)5-s + (0.138 − 0.116i)6-s + (−0.476 − 0.824i)7-s + (−0.176 + 0.306i)8-s + (−0.162 − 0.920i)9-s + (−0.0489 − 0.277i)10-s + (−0.294 + 0.509i)11-s + (−0.0638 − 0.110i)12-s + (1.37 − 1.15i)13-s + (−0.632 + 0.230i)14-s + (0.0957 + 0.0348i)15-s + (0.191 + 0.160i)16-s + (−0.144 + 0.817i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.547367 - 1.29462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.547367 - 1.29462i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.338 - 0.284i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.837 + 0.304i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.26 + 2.18i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.975 - 1.69i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.94 + 4.15i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.594 - 3.37i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (7.69 + 2.79i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.795 + 4.51i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.39 + 7.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + (-2.67 - 2.24i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.86 + 2.13i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.83 + 10.3i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.53 - 1.28i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.493 - 2.79i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.30 - 0.839i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.464 - 2.63i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0530 + 0.0193i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (5.33 + 4.47i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.99 - 5.86i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.06 - 12.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.43 - 1.20i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.35 - 7.70i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−10.08636633952342802172000891007, −9.540547906889641786208887438194, −8.487611591917728703359087704114, −7.67890142601091807252439309389, −6.23380399747922943014003277674, −5.73496646970879692367250532831, −4.00564273671211900520620707321, −3.73161145015562644810047926864, −2.25009747614021370448634817826, −0.68403396353529927426630685813,
1.93904931937691070195284126952, 3.18565080338073607421120945193, 4.45277135505988427332866145416, 5.71530686231451204576187715459, 6.10630863978345020375963781381, 7.21779954697582635133449830314, 8.166754630145171959096744501512, 8.910517710435368819539503278802, 9.566452623987545298927297697691, 10.74849987880521241548256969446
