L(s) = 1 | + (1.57 − 1.57i)3-s + (2.77 − 2.77i)5-s + (2.70 + 2.70i)7-s − 1.94i·9-s + (−3.23 − 3.23i)11-s + 0.941·13-s − 8.73i·15-s + (−3.71 − 1.77i)17-s + 0.880i·19-s + 8.49·21-s + (2.10 + 2.10i)23-s − 10.4i·25-s + (1.66 + 1.66i)27-s + (1.71 − 1.71i)29-s + (−4.36 + 4.36i)31-s + ⋯ |
L(s) = 1 | + (0.907 − 0.907i)3-s + (1.24 − 1.24i)5-s + (1.02 + 1.02i)7-s − 0.647i·9-s + (−0.975 − 0.975i)11-s + 0.261·13-s − 2.25i·15-s + (−0.902 − 0.431i)17-s + 0.202i·19-s + 1.85·21-s + (0.438 + 0.438i)23-s − 2.08i·25-s + (0.320 + 0.320i)27-s + (0.319 − 0.319i)29-s + (−0.784 + 0.784i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.849387803\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.849387803\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.71 + 1.77i)T \) |
good | 3 | \( 1 + (-1.57 + 1.57i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.77 + 2.77i)T - 5iT^{2} \) |
| 7 | \( 1 + (-2.70 - 2.70i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.23 + 3.23i)T + 11iT^{2} \) |
| 13 | \( 1 - 0.941T + 13T^{2} \) |
| 19 | \( 1 - 0.880iT - 19T^{2} \) |
| 23 | \( 1 + (-2.10 - 2.10i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.71 + 1.71i)T - 29iT^{2} \) |
| 31 | \( 1 + (4.36 - 4.36i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.77 - 4.77i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.66iT - 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 6.47iT - 59T^{2} \) |
| 61 | \( 1 + (0.778 + 0.778i)T + 61iT^{2} \) |
| 67 | \( 1 + 7.35T + 67T^{2} \) |
| 71 | \( 1 + (11.2 - 11.2i)T - 71iT^{2} \) |
| 73 | \( 1 + (-10.4 + 10.4i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.22 + 1.22i)T + 79iT^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + (5.49 - 5.49i)T - 97iT^{2} \) |
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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
−9.246217365143677789776701791340, −8.640528527816952859423734052636, −8.423974068860886830951657077821, −7.43750638461069392138472615970, −6.14788865619772962869993888622, −5.39503062778618502742877570542, −4.78503477193738003421214847535, −2.89507222448896274844495010838, −2.09314524569738110749956248932, −1.27965735502629145025152510151,
1.94541627555119106121761660047, 2.66347547221302406324446777418, 3.85756059740550467122531064461, 4.66563566855078854195835382833, 5.70148220571658626070165761629, 6.93934724176184019656491357656, 7.44985934846417195084971095370, 8.564951418477831942083046108036, 9.363294820390034208038806533780, 10.15221057436118656652765183040