L(s) = 1 | + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s − i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s − i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.316358888\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316358888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
good | 3 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.347iT - T^{2} \) |
| 41 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.766 - 0.642i)T^{2} \) |
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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
−8.459424019596921319025735996074, −7.87551381704565072707940793792, −7.34185625306633107385137117118, −6.34433816117544612258188447220, −5.01078077269123625308352239872, −4.63804786731843626211659055779, −3.78629895476868934294499498766, −2.91951261642478425809591809181, −1.65871990825826427361377317461, −1.02978445708095806744248117483,
1.44429319361481778165190947211, 2.04905647114621949797222304320, 3.81961640945775764132732532941, 4.65504221496309316328198481292, 5.05202578907980099472286711061, 5.92465822453296900485343627978, 6.97518770134183356462937722738, 7.36049942651516457237368010889, 7.932739823396552466772707135940, 8.899817632867806001294316819252