| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−0.378 + 0.0823i)3-s + (−0.989 − 0.142i)4-s + (−1.71 + 1.43i)5-s + (0.0551 + 0.383i)6-s + (−2.63 + 1.43i)7-s + (−0.212 + 0.977i)8-s + (−2.59 + 1.18i)9-s + (1.30 + 1.81i)10-s + (−4.70 − 4.07i)11-s + (0.386 − 0.0276i)12-s + (1.13 − 2.08i)13-s + (1.24 + 2.72i)14-s + (0.531 − 0.684i)15-s + (0.959 + 0.281i)16-s + (3.38 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−0.218 + 0.0475i)3-s + (−0.494 − 0.0711i)4-s + (−0.767 + 0.641i)5-s + (0.0225 + 0.156i)6-s + (−0.995 + 0.543i)7-s + (−0.0751 + 0.345i)8-s + (−0.864 + 0.394i)9-s + (0.413 + 0.573i)10-s + (−1.41 − 1.22i)11-s + (0.111 − 0.00797i)12-s + (0.315 − 0.577i)13-s + (0.333 + 0.729i)14-s + (0.137 − 0.176i)15-s + (0.239 + 0.0704i)16-s + (0.821 + 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0637783 + 0.137850i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0637783 + 0.137850i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 5 | \( 1 + (1.71 - 1.43i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
| good | 3 | \( 1 + (0.378 - 0.0823i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (2.63 - 1.43i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (4.70 + 4.07i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.13 + 2.08i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.38 - 2.53i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.894 - 6.22i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (5.28 - 0.759i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.87 - 1.20i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 0.616i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.35 - 7.35i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.416 - 1.91i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (6.39 + 6.39i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.221 + 0.405i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (3.35 + 11.4i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.37 - 5.24i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (14.3 + 1.02i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-0.153 - 0.177i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-6.66 - 8.90i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (9.47 - 2.78i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.55 - 6.83i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (3.96 + 2.55i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.50 - 14.7i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.53921937713601631354619250358, −11.43370853501909037824181496780, −10.73975743356120589223330879220, −10.05442213007613923105723982779, −8.475601956923628356436225633767, −7.963863085049660390209296481349, −6.15978526028399915423064069089, −5.40418051756442541487192837481, −3.44971802159202377243061081489, −2.87747616925750170073137964944,
0.11938569489401101145208009549, 3.21717780396400939245070523176, 4.61570810700880714277991625868, 5.57890350314205920772011324605, 7.02634799686980935484053279823, 7.57958302213178149704769610354, 8.919327696153174461426948727390, 9.642987212987146521416349950202, 10.96193654271963672068469878291, 12.02761697436062268922801352071
