| L(s) = 1 | + (0.676 + 1.24i)2-s + (−0.457 + 0.396i)3-s + (−1.08 + 1.68i)4-s + (−3.79 + 0.545i)5-s + (−0.801 − 0.299i)6-s + (0.480 − 1.05i)7-s + (−2.82 − 0.209i)8-s + (−0.374 + 2.60i)9-s + (−3.24 − 4.34i)10-s + (−1.35 + 4.62i)11-s + (−0.170 − 1.19i)12-s + (5.63 − 2.57i)13-s + (1.63 − 0.115i)14-s + (1.51 − 1.75i)15-s + (−1.64 − 3.64i)16-s + (−0.761 − 0.489i)17-s + ⋯ |
| L(s) = 1 | + (0.478 + 0.878i)2-s + (−0.264 + 0.228i)3-s + (−0.542 + 0.840i)4-s + (−1.69 + 0.243i)5-s + (−0.327 − 0.122i)6-s + (0.181 − 0.398i)7-s + (−0.997 − 0.0740i)8-s + (−0.124 + 0.868i)9-s + (−1.02 − 1.37i)10-s + (−0.409 + 1.39i)11-s + (−0.0491 − 0.346i)12-s + (1.56 − 0.714i)13-s + (0.436 − 0.0308i)14-s + (0.392 − 0.452i)15-s + (−0.412 − 0.911i)16-s + (−0.184 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105350 + 0.831290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105350 + 0.831290i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.676 - 1.24i)T \) |
| 23 | \( 1 + (-0.624 - 4.75i)T \) |
| good | 3 | \( 1 + (0.457 - 0.396i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (3.79 - 0.545i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.480 + 1.05i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.35 - 4.62i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.63 + 2.57i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.761 + 0.489i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.21 - 3.44i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.70 + 4.20i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.49 - 5.19i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.93 + 0.852i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.166 + 1.15i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.41 - 2.09i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 + (-0.959 - 0.438i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 4.94i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.03 - 1.76i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.716 - 2.44i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (12.4 - 3.64i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.67 + 4.28i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.71 + 3.74i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.18 + 0.313i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.22 + 2.56i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.561 + 3.90i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.12876668857889581862118100541, −12.14799474174355880482995720356, −11.26654828630796518011870847309, −10.29113090290501556992371261758, −8.536270348454130375138366254690, −7.73240838459389779924774881437, −7.15068065606652949018689032688, −5.50534323279445311799135872913, −4.40140486572622416563996689411, −3.48580308211737093484764237489,
0.70957508948367012815269321244, 3.26014536342333332867102924585, 4.08389124607167714247513479216, 5.54740060614093242696238960210, 6.76193650713644234146331770992, 8.572995753777576675334235510982, 8.824505091089496846317396370318, 10.81304391405088086872192458039, 11.39213414393808426415047422827, 11.93098263609320303798691877333
