| L(s) = 1 | + (0.947 − 1.05i)2-s + (−0.939 + 0.342i)3-s + (−0.205 − 1.98i)4-s + (0.234 − 1.33i)5-s + (−0.530 + 1.31i)6-s + (1.38 − 0.798i)7-s + (−2.28 − 1.66i)8-s + (0.766 − 0.642i)9-s + (−1.17 − 1.50i)10-s + (−2.03 − 1.17i)11-s + (0.873 + 1.79i)12-s + (0.358 − 0.983i)13-s + (0.471 − 2.20i)14-s + (0.234 + 1.33i)15-s + (−3.91 + 0.819i)16-s + (2.41 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.669 − 0.742i)2-s + (−0.542 + 0.197i)3-s + (−0.102 − 0.994i)4-s + (0.104 − 0.594i)5-s + (−0.216 + 0.535i)6-s + (0.522 − 0.301i)7-s + (−0.807 − 0.589i)8-s + (0.255 − 0.214i)9-s + (−0.371 − 0.476i)10-s + (−0.614 − 0.355i)11-s + (0.252 + 0.519i)12-s + (0.0992 − 0.272i)13-s + (0.125 − 0.590i)14-s + (0.0605 + 0.343i)15-s + (−0.978 + 0.204i)16-s + (0.585 + 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.882498 - 1.14449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882498 - 1.14449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.947 + 1.05i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-2.22 + 3.74i)T \) |
| good | 5 | \( 1 + (-0.234 + 1.33i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 0.798i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.358 + 0.983i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.41 - 2.02i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (3.00 - 0.530i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.87 - 2.23i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.06 - 8.77i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.03iT - 37T^{2} \) |
| 41 | \( 1 + (-0.940 - 2.58i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.34 - 1.64i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.10 + 4.88i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.12 - 0.375i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (5.42 + 4.55i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.92 + 10.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.47 - 2.07i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0247 + 0.140i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.09 + 0.762i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.69 + 2.80i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.01 + 3.47i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.39 + 6.56i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (6.02 - 7.18i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−12.01351882979967056620120656427, −10.98904766067787752936751219754, −10.38914836174159373735673202844, −9.293052675941227940314828492415, −8.067481108147108020147486007782, −6.49264462749924779933038965185, −5.28535767817485103075220009491, −4.67159627944732633823823640964, −3.15904849814800932982768436426, −1.18068813995711621585311028722,
2.56873662716317587372250597348, 4.23351687057761783047560338871, 5.42122570935344993181545166561, 6.24068062329631116072466169842, 7.41689739402520478208375103035, 8.091763602501636142800482703288, 9.592435252737956337271855091324, 10.79595030622286768030845463837, 11.83039956063136505054047948440, 12.42577009448699894654969296963
