L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (−3.15 − 1.82i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (1.44 − 0.834i)11-s + (−2.24 − 2.82i)13-s + 3.64·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (5.46 + 3.15i)19-s + (0.866 + 0.499i)20-s + (−0.834 + 1.44i)22-s + (−0.622 − 1.07i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−1.19 − 0.688i)7-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (0.435 − 0.251i)11-s + (−0.622 − 0.782i)13-s + 0.974·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (1.25 + 0.724i)19-s + (0.193 + 0.111i)20-s + (−0.177 + 0.307i)22-s + (−0.129 − 0.224i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7524616409\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7524616409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (2.24 + 2.82i)T \) |
good | 7 | \( 1 + (3.15 + 1.82i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.44 + 0.834i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.46 - 3.15i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.622 + 1.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.02 - 8.69i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.21iT - 31T^{2} \) |
| 37 | \( 1 + (-8.54 + 4.93i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.04 - 4.64i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.78 - 6.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.82iT - 47T^{2} \) |
| 53 | \( 1 - 0.848T + 53T^{2} \) |
| 59 | \( 1 + (5.29 + 3.05i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 7.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 1.75i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 + 7.95iT - 83T^{2} \) |
| 89 | \( 1 + (5.15 - 2.97i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.38 + 1.37i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.983212482059529459727613500016, −9.329575028646096312554309972683, −8.281673633569295280627074591187, −7.51408432699097812551568733171, −6.67887994815967496265537929251, −6.17088990265602561046042404610, −5.01806524897364059901239182919, −3.64067014981547014206668391200, −2.88885764529529089628864547611, −1.15644177063562560081303099987,
0.45049431612109964127777416393, 2.14888817048056919264488047327, 3.02839691061777457433186930859, 4.23630950516548655963989687817, 5.27624364161932029296277901433, 6.45240536453359802314455984372, 7.02807502966216299094755885370, 8.078605030851795887713638811057, 9.046412490988261217614692628774, 9.623851239965509574765566333746