| L(s) = 1 | + (−1 − 1.41i)3-s + 1.41i·5-s + 4.24i·7-s + (−1.00 + 2.82i)9-s − 13-s + (2.00 − 1.41i)15-s − 5.65i·17-s + 4.24i·19-s + (6 − 4.24i)21-s + 6·23-s + 2.99·25-s + (5.00 − 1.41i)27-s − 2.82i·29-s + 4.24i·31-s − 6·35-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s + 0.632i·5-s + 1.60i·7-s + (−0.333 + 0.942i)9-s − 0.277·13-s + (0.516 − 0.365i)15-s − 1.37i·17-s + 0.973i·19-s + (1.30 − 0.925i)21-s + 1.25·23-s + 0.599·25-s + (0.962 − 0.272i)27-s − 0.525i·29-s + 0.762i·31-s − 1.01·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8460148157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8460148157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.115927017210107657419112042440, −8.359434629217760221147641005071, −7.55148289238900479761853146416, −6.79676077873873158056216667873, −6.17844835843997185499768170738, −5.34876883963515057545429861033, −4.81357870413969561667446293811, −3.03500125608773468115022531176, −2.59951624081907006934421405271, −1.41819956229099312237092660478,
0.32592372835047223712539166839, 1.38065508868933377644396312672, 3.17779129694635953876009540049, 3.97426791652236557253659632781, 4.70066714191072884606691469290, 5.23117870242481600587779957832, 6.42372197197166788844178362495, 6.96948567783300939658450681801, 7.920460628607853711662591199969, 8.839088466692589603269792597626
