L(s) = 1 | + (1 + 1.73i)5-s + (−2 + 3.46i)11-s + 2·13-s + (−3 + 5.19i)17-s + (−4 − 6.92i)19-s + (0.500 − 0.866i)25-s − 6·29-s + (−4 + 6.92i)31-s + (1 + 1.73i)37-s − 2·41-s − 4·43-s + (−4 − 6.92i)47-s + (3 − 5.19i)53-s − 7.99·55-s + (3 + 5.19i)61-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (−0.603 + 1.04i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (−0.917 − 1.58i)19-s + (0.100 − 0.173i)25-s − 1.11·29-s + (−0.718 + 1.24i)31-s + (0.164 + 0.284i)37-s − 0.312·41-s − 0.609·43-s + (−0.583 − 1.01i)47-s + (0.412 − 0.713i)53-s − 1.07·55-s + (0.384 + 0.665i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5010744197\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5010744197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.788359837733482487556373952670, −8.407017069926959385513442878512, −7.18966060393141447416276885459, −6.83686683140558046296564218602, −6.10721642595436179116271076953, −5.15880243154139159184087563759, −4.38682891005626220835059926523, −3.44222563005466661107446109977, −2.40472872855139262739252710613, −1.77459012431860518593929449632,
0.13794029871789830177843569088, 1.42328680425892955404019516207, 2.41984590654836277065955255392, 3.51889863051200712724345189545, 4.31809853736027382662252033508, 5.36124022439734986559795617703, 5.75636346312353936597940121682, 6.57266779395104694305806105163, 7.62135792691190498769032676722, 8.242792853811672089924177324000