| L(s) = 1 | + (2.88 + 0.252i)2-s + (−5.10 + 0.975i)3-s + (0.363 + 0.0640i)4-s + (9.52 − 5.85i)5-s + (−14.9 + 1.52i)6-s + (−5.91 − 4.14i)7-s + (−21.3 − 5.71i)8-s + (25.0 − 9.95i)9-s + (28.9 − 14.4i)10-s + (8.48 − 23.3i)11-s + (−1.91 + 0.0272i)12-s + (−7.43 − 84.9i)13-s + (−15.9 − 13.4i)14-s + (−42.8 + 39.1i)15-s + (−62.7 − 22.8i)16-s + (107. − 28.8i)17-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.0891i)2-s + (−0.982 + 0.187i)3-s + (0.0453 + 0.00800i)4-s + (0.851 − 0.523i)5-s + (−1.01 + 0.103i)6-s + (−0.319 − 0.223i)7-s + (−0.942 − 0.252i)8-s + (0.929 − 0.368i)9-s + (0.914 − 0.457i)10-s + (0.232 − 0.639i)11-s + (−0.0460 + 0.000655i)12-s + (−0.158 − 1.81i)13-s + (−0.305 − 0.256i)14-s + (−0.738 + 0.674i)15-s + (−0.980 − 0.357i)16-s + (1.53 − 0.411i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.22468 - 1.10942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22468 - 1.10942i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.10 - 0.975i)T \) |
| 5 | \( 1 + (-9.52 + 5.85i)T \) |
| good | 2 | \( 1 + (-2.88 - 0.252i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (5.91 + 4.14i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-8.48 + 23.3i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (7.43 + 84.9i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-107. + 28.8i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (109. - 63.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (20.0 + 28.6i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-47.8 + 40.1i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-4.35 + 24.6i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-51.5 - 192. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (292. - 349. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-87.2 - 187. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (-167. + 239. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (228. - 228. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-219. + 79.7i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-33.7 - 191. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-947. + 82.9i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (251. + 145. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-188. + 702. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (182. + 217. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (8.21 - 93.8i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (43.7 + 75.8i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.40e3 + 656. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.72939387615187027486025521478, −11.90356240211148672095857499126, −10.35512386129086950575002418409, −9.795459771527498754924624456887, −8.236293059372652358233805940219, −6.32299348321478955320855974834, −5.70619510601226151866336460772, −4.81649467720351571604115805930, −3.35110238101174818338791560180, −0.69231224935429370540236114197,
2.07677318116928421479855463291, 4.00093200698522984128871525272, 5.19040222324693034741290035222, 6.22215473672895391875973889021, 6.97428555453817419758646230261, 9.086028155112547029990751557947, 10.04165211710609584783510476966, 11.25326287902812284038917149871, 12.25538735054184170189320767057, 12.83262131309827859373271045615
