| L(s) = 1 | + (0.118 + 0.0570i)2-s + (1.09 + 0.744i)3-s + (−1.23 − 1.55i)4-s + (−1.30 + 1.21i)5-s + (0.0867 + 0.150i)6-s + (−0.00749 + 0.0129i)7-s + (−0.116 − 0.510i)8-s + (−0.458 − 1.16i)9-s + (−0.223 + 0.0689i)10-s + (−1.29 + 1.62i)11-s + (−0.195 − 2.61i)12-s + (−1.55 − 0.478i)13-s + (−0.00162 + 0.00110i)14-s + (−2.32 + 0.350i)15-s + (−0.867 + 3.79i)16-s + (4.42 + 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.0837 + 0.0403i)2-s + (0.630 + 0.429i)3-s + (−0.618 − 0.775i)4-s + (−0.583 + 0.541i)5-s + (0.0354 + 0.0613i)6-s + (−0.00283 + 0.00490i)7-s + (−0.0411 − 0.180i)8-s + (−0.152 − 0.389i)9-s + (−0.0707 + 0.0218i)10-s + (−0.390 + 0.490i)11-s + (−0.0564 − 0.753i)12-s + (−0.430 − 0.132i)13-s + (−0.000434 + 0.000296i)14-s + (−0.600 + 0.0905i)15-s + (−0.216 + 0.949i)16-s + (1.07 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.806257 + 0.0511136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.806257 + 0.0511136i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.84 + 6.29i)T \) |
| good | 2 | \( 1 + (-0.118 - 0.0570i)T + (1.24 + 1.56i)T^{2} \) |
| 3 | \( 1 + (-1.09 - 0.744i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (1.30 - 1.21i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (0.00749 - 0.0129i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.29 - 1.62i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.55 + 0.478i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-4.42 - 4.10i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.78 + 7.09i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (-5.24 - 0.790i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (5.92 - 4.04i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.178 - 2.37i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (2.52 + 4.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.20 + 1.54i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (6.24 + 7.83i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (6.55 - 2.02i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.370 + 1.62i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.452 - 6.03i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-0.523 + 1.33i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (-6.96 + 1.05i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-9.32 - 2.87i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (6.00 - 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.49 - 5.10i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (13.1 + 8.95i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-8.44 + 10.5i)T + (-21.5 - 94.5i)T^{2} \) |
| 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
−15.34587512911510912696454744443, −15.08537591799503166698948778187, −14.04565508558853176323271168477, −12.69154514403188590337864853516, −11.04669653743494269512385680742, −9.854842940417090532478385888934, −8.825667740412704769491563348694, −7.16582627639094492249646541834, −5.20809056694152857441552959087, −3.47651686551787057643016413717,
3.25274293146344409606591584418, 5.06166594297125518714009833640, 7.67993629115187433337468156032, 8.195686421005647971820426372508, 9.598031866094775375674294594350, 11.60713195499497807004302887116, 12.61519510459172113795049574619, 13.60704093189868794593026513620, 14.52562085865681315111386450794, 16.28610481168611174849308195662
