| L(s) = 1 | + (0.991 − 0.130i)2-s + (0.442 + 0.896i)3-s + (0.965 − 0.258i)4-s + (2.50 − 2.20i)5-s + (0.555 + 0.831i)6-s + (2.62 + 0.313i)7-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (2.20 − 2.50i)10-s + (−4.53 + 0.297i)11-s + (0.659 + 0.751i)12-s + (3.69 + 3.69i)13-s + (2.64 − 0.0318i)14-s + (3.08 + 1.27i)15-s + (0.866 − 0.5i)16-s + (−3.44 − 2.25i)17-s + ⋯ |
| L(s) = 1 | + (0.701 − 0.0922i)2-s + (0.255 + 0.517i)3-s + (0.482 − 0.129i)4-s + (1.12 − 0.984i)5-s + (0.226 + 0.339i)6-s + (0.992 + 0.118i)7-s + (0.326 − 0.135i)8-s + (−0.202 + 0.264i)9-s + (0.695 − 0.793i)10-s + (−1.36 + 0.0896i)11-s + (0.190 + 0.217i)12-s + (1.02 + 1.02i)13-s + (0.707 − 0.00851i)14-s + (0.796 + 0.329i)15-s + (0.216 − 0.125i)16-s + (−0.836 − 0.547i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.08098 - 0.142717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.08098 - 0.142717i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 3 | \( 1 + (-0.442 - 0.896i)T \) |
| 7 | \( 1 + (-2.62 - 0.313i)T \) |
| 17 | \( 1 + (3.44 + 2.25i)T \) |
| good | 5 | \( 1 + (-2.50 + 2.20i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (4.53 - 0.297i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-3.69 - 3.69i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.396 + 3.01i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (3.60 + 1.77i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.811 - 4.07i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (1.96 - 0.968i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.0859 + 1.31i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (1.02 - 5.16i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.09 + 5.06i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (1.18 - 4.40i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.13 - 6.69i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (12.5 + 1.65i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.638 + 1.88i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-5.13 - 2.96i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.20 - 1.47i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-14.8 - 5.03i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-1.88 + 3.82i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-6.46 + 15.6i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (7.07 + 1.89i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (18.3 - 3.65i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.61734018537035749484675324208, −9.452027193498645068080357482312, −8.816443745274044935606288705899, −7.997651201696790266923691038174, −6.64909053415086693609727616034, −5.54377866721786203089267033875, −4.94444516556368325755533917943, −4.24830499165593068427343501099, −2.58165480098033156740063285626, −1.66372606376068409957177793794,
1.81038160406517972699162420788, 2.61154206291318561631163262302, 3.78721431760404436598204660700, 5.30873022345049766695865554586, 5.90047687661609049614389703538, 6.73081661586922077336636917887, 7.913713647495557989764526057108, 8.264811513801348875969410497970, 9.832660482039530400438469622949, 10.74906050548630684947774716824
