| L(s) = 1 | + (−1.39 + 0.234i)2-s + (−0.467 + 0.249i)3-s + (1.89 − 0.652i)4-s + (0.246 + 0.300i)5-s + (0.592 − 0.457i)6-s + (1.89 − 1.26i)7-s + (−2.48 + 1.35i)8-s + (−1.51 + 2.26i)9-s + (−0.413 − 0.360i)10-s + (5.42 − 1.64i)11-s + (−0.719 + 0.776i)12-s + (5.51 + 4.52i)13-s + (−2.34 + 2.20i)14-s + (−0.189 − 0.0786i)15-s + (3.14 − 2.46i)16-s + (−3.03 + 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 + 0.165i)2-s + (−0.269 + 0.144i)3-s + (0.945 − 0.326i)4-s + (0.110 + 0.134i)5-s + (0.242 − 0.186i)6-s + (0.714 − 0.477i)7-s + (−0.878 + 0.478i)8-s + (−0.503 + 0.753i)9-s + (−0.130 − 0.114i)10-s + (1.63 − 0.496i)11-s + (−0.207 + 0.224i)12-s + (1.53 + 1.25i)13-s + (−0.626 + 0.589i)14-s + (−0.0490 − 0.0203i)15-s + (0.786 − 0.617i)16-s + (−0.735 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.730833 + 0.158657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.730833 + 0.158657i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.234i)T \) |
| good | 3 | \( 1 + (0.467 - 0.249i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (-0.246 - 0.300i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.89 + 1.26i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-5.42 + 1.64i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-5.51 - 4.52i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (3.03 - 1.25i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.249 + 2.53i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (5.91 - 1.17i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.110 - 0.363i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (0.158 + 0.158i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.28 + 0.421i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.22 + 6.16i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (7.59 + 4.05i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.45 - 3.51i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 6.80i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-7.12 + 5.84i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.08 - 3.90i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-1.02 - 1.92i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (2.79 + 4.17i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.95 + 5.98i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (3.96 - 9.56i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (10.9 - 1.07i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (2.99 + 0.594i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.90 - 4.90i)T + 97iT^{2} \) |
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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
−13.87676035797061439008311629414, −11.73609099908029092854648285845, −11.28763835306928978007981992175, −10.43402568831618375817870547016, −8.948765791855461475748345388521, −8.402823494753168485200977969698, −6.85793658125283216760677701218, −6.00583175133307935631928186406, −4.11297057559539101635216587781, −1.71691573623690028525774912696,
1.47461184042137867245559421968, 3.59764317880156887118320596109, 5.80873256170332533981271332578, 6.72103665416507808860931298865, 8.306668263128676879168541585002, 8.898133975658791977502918209990, 10.08410590125043624958720347087, 11.43249923432255753215554097534, 11.77715659218926061848413565583, 12.98125843922950647499468540961
