L(s) = 1 | + 3-s − 5-s + (−0.550 − 0.952i)7-s + 9-s + (−0.589 − 1.02i)11-s + (0.5 − 0.866i)13-s − 15-s + (−1.83 + 3.17i)17-s + (−3.26 + 5.65i)19-s + (−0.550 − 0.952i)21-s + (4.11 − 7.12i)23-s + 25-s + 27-s + (4.58 + 7.94i)29-s + (1.12 + 1.94i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + (−0.207 − 0.360i)7-s + 0.333·9-s + (−0.177 − 0.307i)11-s + (0.138 − 0.240i)13-s − 0.258·15-s + (−0.443 + 0.768i)17-s + (−0.748 + 1.29i)19-s + (−0.120 − 0.207i)21-s + (0.857 − 1.48i)23-s + 0.200·25-s + 0.192·27-s + (0.851 + 1.47i)29-s + (0.201 + 0.349i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.894603304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.894603304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-8.12 + 0.997i)T \) |
good | 7 | \( 1 + (0.550 + 0.952i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.589 + 1.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.26 - 5.65i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.11 + 7.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.58 - 7.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 1.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.58 - 9.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.73 + 8.20i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + (-5.16 - 8.94i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 - 8.73T + 59T^{2} \) |
| 61 | \( 1 + (-7.40 + 12.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.07 - 5.32i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.45 - 7.71i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.22 + 2.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.48 - 9.50i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.19T + 89T^{2} \) |
| 97 | \( 1 + (-3.18 + 5.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.492713489702654093540207135726, −8.035616814332804518646415596293, −6.85881055671282205890082226719, −6.64976927100535310846253657774, −5.48203199004891723013568625345, −4.59776957551658394308941262571, −3.79883333888249978928818486251, −3.16910718890180217389467494963, −2.13440519239245292889334571323, −0.935880450309957144400092918412,
0.62368182913077868489132600036, 2.14783630828241307783737962103, 2.76841927859448179166847158252, 3.76591613120541798150323915225, 4.52409239005001139306440205763, 5.28535943989077789326053097964, 6.28793663674368628169565124491, 7.16362327262587199711070115987, 7.48664059975276917486052336011, 8.612194878123555016451889023891