| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.121 − 0.841i)5-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.353 − 0.773i)10-s + (2.81 + 0.826i)11-s + (0.959 + 0.281i)12-s + (2.09 + 4.58i)13-s + (−0.142 + 0.989i)14-s + (0.556 − 0.642i)15-s + (0.415 − 0.909i)16-s + (−2.26 − 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0541 − 0.376i)5-s + (0.343 + 0.220i)6-s + (−0.157 + 0.343i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.111 − 0.244i)10-s + (0.849 + 0.249i)11-s + (0.276 + 0.0813i)12-s + (0.580 + 1.27i)13-s + (−0.0380 + 0.264i)14-s + (0.143 − 0.165i)15-s + (0.103 − 0.227i)16-s + (−0.548 − 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.81506 + 0.347213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.81506 + 0.347213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.66 - 3.98i)T \) |
| good | 5 | \( 1 + (0.121 + 0.841i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-2.81 - 0.826i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.09 - 4.58i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.26 + 1.45i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.58 + 2.94i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.33 - 2.14i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.386 - 0.446i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.38 + 9.63i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.105 - 0.733i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.38 - 1.59i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 0.155T + 47T^{2} \) |
| 53 | \( 1 + (-3.04 + 6.65i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (3.93 + 8.62i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.79 - 10.1i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-7.41 + 2.17i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (14.8 - 4.36i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (13.4 - 8.67i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.65 - 10.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.213 - 1.48i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (8.10 + 9.35i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.138 + 0.959i)T + (-93.0 + 27.3i)T^{2} \) |
| 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
−9.929077503678898846228047060502, −9.174425970428449179160768844398, −8.730817698713069502481313748872, −7.34196628813532478997199839772, −6.59641664179926241954540944375, −5.54788807959025161475549644612, −4.56952596395015723240642898525, −3.90050789097310347178889871810, −2.78151654199892474937894423673, −1.51500095362195509010747592533,
1.24063095228728714530167274934, 2.84690330377638852718635624199, 3.53745223734896617343037119583, 4.59188943263120870914548575669, 5.93203050762064475787475614317, 6.43130933044547982018042680825, 7.40992340820808887140336986602, 8.136587040558945098538199680811, 8.966589598786131205536038727156, 10.17579618276342128388851811236
