L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.766 − 0.642i)6-s + (−0.326 + 0.565i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (−1.43 − 2.49i)11-s + (−0.499 + 0.866i)12-s + (−1.26 − 1.06i)13-s + (0.613 + 0.223i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (−1.26 − 7.18i)17-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (−0.312 − 0.262i)6-s + (−0.123 + 0.213i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (−0.0549 + 0.311i)10-s + (−0.434 − 0.751i)11-s + (−0.144 + 0.249i)12-s + (−0.351 − 0.294i)13-s + (0.163 + 0.0596i)14-s + (−0.242 + 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.307 − 1.74i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0900416 - 0.907921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0900416 - 0.907921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (3.79 + 2.15i)T \) |
good | 7 | \( 1 + (0.326 - 0.565i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.43 + 2.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.26 + 1.06i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.26 + 7.18i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.61 - 0.587i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0282 + 0.160i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.471 - 0.817i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + (0.309 - 0.259i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.6 - 3.88i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.00 + 5.71i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-9.93 + 3.61i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.790 + 4.48i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (10.8 - 3.93i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.396 - 2.24i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.01 - 1.45i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.73 + 7.32i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (3.73 - 3.13i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.14 + 3.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.91 - 1.60i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.43 + 13.8i)T + (-91.1 + 33.1i)T^{2} \) |
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
−10.42376007026106353171240187476, −9.313152913206409961394689287714, −8.727910629912058946147212059695, −7.80367573842345366745589209363, −6.93975622202683208307852400709, −5.55895120630565437815067054357, −4.47390220807510607011636566771, −3.20653929455301551457809703071, −2.35082057719640196427356398328, −0.49505754036901851570277907216,
2.11261080324893219195223820529, 3.80231476069122600728715484167, 4.44795998934861232981555917942, 5.73416648119070447452539245585, 6.78012022532417434690554365842, 7.66595791552742188101942809197, 8.407218127620115830470889600893, 9.233598599858421587123636512218, 10.35485305729759500773815961815, 10.67416349535965730606350945119