L(s) = 1 | + 2-s + 4-s − 2.50i·5-s + i·7-s + 8-s − 2.50i·10-s + (3.23 + 0.715i)11-s − 1.06i·13-s + i·14-s + 16-s + 1.11·17-s − 0.185i·19-s − 2.50i·20-s + (3.23 + 0.715i)22-s − 5.70i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.11i·5-s + 0.377i·7-s + 0.353·8-s − 0.791i·10-s + (0.976 + 0.215i)11-s − 0.296i·13-s + 0.267i·14-s + 0.250·16-s + 0.270·17-s − 0.0425i·19-s − 0.559i·20-s + (0.690 + 0.152i)22-s − 1.18i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.773255807\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.773255807\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-3.23 - 0.715i)T \) |
good | 5 | \( 1 + 2.50iT - 5T^{2} \) |
| 13 | \( 1 + 1.06iT - 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 0.185iT - 19T^{2} \) |
| 23 | \( 1 + 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 + 0.132T + 41T^{2} \) |
| 43 | \( 1 + 8.83iT - 43T^{2} \) |
| 47 | \( 1 - 5.08iT - 47T^{2} \) |
| 53 | \( 1 + 0.0980iT - 53T^{2} \) |
| 59 | \( 1 + 6.33iT - 59T^{2} \) |
| 61 | \( 1 + 3.93iT - 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 + 0.698iT - 71T^{2} \) |
| 73 | \( 1 - 0.169iT - 73T^{2} \) |
| 79 | \( 1 + 1.63iT - 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 2.71T + 97T^{2} \) |
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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
−9.307157801263642552005327235893, −8.728709176293133055547333869107, −7.904511670222464645756866092138, −6.82521907349148086744936181198, −6.05455294872112099817158041871, −5.12685157333167091943265165258, −4.49683564947634761629376024275, −3.54299921277061862048654928477, −2.26775582197195611266641915576, −1.01222236290596235871014741772,
1.47679540564015785221400563905, 2.84074622584097552354550377142, 3.60031711584267643987027078669, 4.43621934016848694345234294462, 5.63440553301225897107952621065, 6.43764412264352887131052738815, 7.05548820435952292252932961057, 7.76830701109413979471135916798, 8.948655271119813875271077780045, 9.840999267208315660208107302569