L(s) = 1 | + 2-s + (1.68 + 0.396i)3-s + 4-s + (2.18 + 0.469i)5-s + (1.68 + 0.396i)6-s + 8-s + (2.68 + 1.33i)9-s + (2.18 + 0.469i)10-s − 0.939i·11-s + (1.68 + 0.396i)12-s − 2·13-s + (3.5 + 1.65i)15-s + 16-s − 6.63i·17-s + (2.68 + 1.33i)18-s + 3.46i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.973 + 0.228i)3-s + 0.5·4-s + (0.977 + 0.210i)5-s + (0.688 + 0.161i)6-s + 0.353·8-s + (0.895 + 0.445i)9-s + (0.691 + 0.148i)10-s − 0.283i·11-s + (0.486 + 0.114i)12-s − 0.554·13-s + (0.903 + 0.428i)15-s + 0.250·16-s − 1.60i·17-s + (0.633 + 0.314i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.344142632\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.344142632\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.68 - 0.396i)T \) |
| 5 | \( 1 + (-2.18 - 0.469i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.939iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 3.31iT - 29T^{2} \) |
| 31 | \( 1 - 7.57iT - 31T^{2} \) |
| 37 | \( 1 + 8.21iT - 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 1.08iT - 43T^{2} \) |
| 47 | \( 1 - 8.51iT - 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 2.37iT - 67T^{2} \) |
| 71 | \( 1 + 8.51iT - 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 11.8iT - 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−9.527583321076411132136750968339, −8.920214243751477386348283802536, −7.78806388417301537499593467344, −7.14250799121799863992095770651, −6.21064052980526673778306437603, −5.24754836124602844190963063364, −4.52143596600982870829932916128, −3.29645867635647594803691254247, −2.65043453865690976503425475067, −1.63508620236438126454758752304,
1.57396666702928699457865643220, 2.32272535519652762630525380616, 3.34570751385865511769093004358, 4.37174184590188199655129735067, 5.21986489062575304304215038602, 6.30597095849735591005708759598, 6.85387289169299466045722712391, 7.924364400790202836902206858236, 8.601245059663153955906298229895, 9.555099353556276588851484872464