| L(s) = 1 | + (1.61 − 1.17i)2-s + (0.168 + 0.519i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (0.884 + 0.642i)6-s + (3.58 − 11.0i)7-s + (−2.47 − 7.60i)8-s + (21.6 − 15.6i)9-s − 10·10-s + (−21.7 − 29.2i)11-s + 2.18·12-s + (29.3 − 21.2i)13-s + (−7.17 − 22.0i)14-s + (0.844 − 2.59i)15-s + (−12.9 − 9.40i)16-s + (−19.2 − 13.9i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.0324 + 0.100i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.0601 + 0.0437i)6-s + (0.193 − 0.596i)7-s + (−0.109 − 0.336i)8-s + (0.800 − 0.581i)9-s − 0.316·10-s + (−0.597 − 0.802i)11-s + 0.0525·12-s + (0.625 − 0.454i)13-s + (−0.136 − 0.421i)14-s + (0.0145 − 0.0447i)15-s + (−0.202 − 0.146i)16-s + (−0.274 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.43994 - 1.45739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43994 - 1.45739i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 5 | \( 1 + (4.04 + 2.93i)T \) |
| 11 | \( 1 + (21.7 + 29.2i)T \) |
| good | 3 | \( 1 + (-0.168 - 0.519i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + (-3.58 + 11.0i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-29.3 + 21.2i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (19.2 + 13.9i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-20.4 - 63.0i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 23.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + (16.4 - 50.5i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-32.3 + 23.5i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (72.7 - 223. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-58.7 - 180. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-22.5 - 69.4i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-397. + 288. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (187. - 575. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-584. - 424. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 380.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (350. + 254. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-313. + 965. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (898. - 653. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (51.6 + 37.5i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 286.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.40e3 + 1.02e3i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.01193267990423189540588388921, −11.94002744289388030297442726763, −10.88062917687161907319962108392, −10.01766648027642211859740663603, −8.582208619162298612993883561935, −7.30194962789793316973895365166, −5.86205970168325635113806958967, −4.43446581997911994998657846088, −3.31186025463679516627737081860, −1.03061030836672758833808789726,
2.28477059614129082723958796945, 4.13245643412980690067736971254, 5.30105747326989967741202805519, 6.81478641552940006584448400217, 7.69918829350942413260746260853, 8.947388552277811839107734908864, 10.43798963940556020722775078793, 11.51956235615621018811642980621, 12.60888651477231969929215127720, 13.39683803828297146764994460147
