| L(s) = 1 | + (−0.959 + 0.281i)2-s + (2.00 + 2.31i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−2.57 − 1.65i)6-s + (0.414 − 0.908i)7-s + (−0.654 + 0.755i)8-s + (−0.911 + 6.33i)9-s + (−0.415 − 0.909i)10-s + (4.67 + 1.37i)11-s + (2.94 + 0.863i)12-s + (−2.42 − 5.31i)13-s + (−0.142 + 0.988i)14-s + (−2.00 + 2.31i)15-s + (0.415 − 0.909i)16-s + (−3.46 − 2.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (1.15 + 1.33i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (−1.05 − 0.676i)6-s + (0.156 − 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.303 + 2.11i)9-s + (−0.131 − 0.287i)10-s + (1.40 + 0.413i)11-s + (0.849 + 0.249i)12-s + (−0.672 − 1.47i)13-s + (−0.0379 + 0.264i)14-s + (−0.518 + 0.598i)15-s + (0.103 − 0.227i)16-s + (−0.841 − 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0191 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0191 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939234 + 0.921387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939234 + 0.921387i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-4.02 + 2.60i)T \) |
| good | 3 | \( 1 + (-2.00 - 2.31i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.414 + 0.908i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.67 - 1.37i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.42 + 5.31i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (3.46 + 2.22i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (5.79 - 3.72i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.04 - 1.95i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 1.26i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.00888 - 0.0617i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.409 + 2.85i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.13 + 5.92i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 + (1.43 - 3.13i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.84 + 10.6i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.283 - 0.327i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.86 + 1.42i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.68 + 2.25i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.16 + 5.88i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (6.44 + 14.1i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.929 - 6.46i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.28 - 4.94i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.140 - 0.975i)T + (-93.0 + 27.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.36062221628998353239043683710, −10.83961404500358767826304884502, −10.38801712864058761934116522949, −9.468493480205722572485966842566, −8.725645531131754930814782729238, −7.79621656772681934369878196304, −6.60747664882205846240538963167, −4.88127475170892496283312962369, −3.73154469617027370372715942058, −2.43130041090557837300228447551,
1.46132703955722229297070711801, 2.48624835695713101375408008571, 4.16581185943663003674954314187, 6.52159286438072720625732416777, 6.92440679482247516252244113551, 8.344429560660184727730091917160, 8.870613589239475732060586490290, 9.451608302188099440287604267002, 11.29839946793417744956664114754, 12.00435762573121963402287703654
