L(s) = 1 | + (0.655 + 1.36i)2-s + (−1.90 − 0.435i)3-s + (−0.176 + 0.220i)4-s + (−1.27 − 1.83i)5-s + (−0.657 − 2.87i)6-s + (2.62 + 0.335i)7-s + (2.52 + 0.577i)8-s + (0.740 + 0.356i)9-s + (1.66 − 2.93i)10-s + (3.72 − 1.79i)11-s + (0.431 − 0.344i)12-s + (−0.574 − 1.19i)13-s + (1.26 + 3.79i)14-s + (1.62 + 4.05i)15-s + (0.998 + 4.37i)16-s + (3.02 − 2.40i)17-s + ⋯ |
L(s) = 1 | + (0.463 + 0.962i)2-s + (−1.10 − 0.251i)3-s + (−0.0880 + 0.110i)4-s + (−0.569 − 0.822i)5-s + (−0.268 − 1.17i)6-s + (0.991 + 0.126i)7-s + (0.894 + 0.204i)8-s + (0.246 + 0.118i)9-s + (0.527 − 0.928i)10-s + (1.12 − 0.540i)11-s + (0.124 − 0.0993i)12-s + (−0.159 − 0.330i)13-s + (0.337 + 1.01i)14-s + (0.419 + 1.04i)15-s + (0.249 + 1.09i)16-s + (0.732 − 0.584i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24691 + 0.0393128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24691 + 0.0393128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.27 + 1.83i)T \) |
| 7 | \( 1 + (-2.62 - 0.335i)T \) |
good | 2 | \( 1 + (-0.655 - 1.36i)T + (-1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (1.90 + 0.435i)T + (2.70 + 1.30i)T^{2} \) |
| 11 | \( 1 + (-3.72 + 1.79i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.574 + 1.19i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 2.40i)T + (3.78 - 16.5i)T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + (6.09 + 4.86i)T + (5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 1.54i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 + (1.76 - 1.40i)T + (8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + (1.10 - 4.86i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-6.43 + 1.46i)T + (38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (2.18 + 4.53i)T + (-29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (1.73 + 1.38i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-0.454 - 1.98i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-7.05 - 8.84i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 8.42iT - 67T^{2} \) |
| 71 | \( 1 + (-0.457 + 0.573i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.13 - 12.7i)T + (-45.5 - 57.0i)T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + (-6.15 + 12.7i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.448 + 0.215i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 - 9.93iT - 97T^{2} \) |
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
−11.83538290757329907015314017963, −11.60596546702087484144672502571, −10.45212661253274677087991910040, −8.853299683773377049638783239977, −7.901898466021805887378851648326, −6.94645166087131032700997454311, −5.76116944866552475964320701978, −5.23364891427112786210269089472, −4.13768568287575915325004000541, −1.18593696501857889977175961697,
1.78898306382043160187863689845, 3.62971443312443799617884393165, 4.44822236987415668118453564204, 5.73282787544333805250026060345, 7.07949211097324758364599253745, 7.921807292020106557353644646047, 9.718743538692323627150626273415, 10.70564363944879893918520003029, 11.28020177119602505116985500559, 11.89838329185685510475135227459