L(s) = 1 | + (−2.97 + 1.23i)5-s + (−0.237 − 0.237i)7-s + (2.12 − 0.879i)11-s + (0.0390 − 0.0943i)13-s − 4.16·17-s + (4.25 + 1.76i)19-s + (−4.84 − 4.84i)23-s + (3.79 − 3.79i)25-s + (−2.90 + 7.01i)29-s − 9.88i·31-s + (0.999 + 0.414i)35-s + (0.175 + 0.424i)37-s + (7.67 − 7.67i)41-s + (−2.99 − 7.23i)43-s − 6.10i·47-s + ⋯ |
L(s) = 1 | + (−1.33 + 0.550i)5-s + (−0.0898 − 0.0898i)7-s + (0.640 − 0.265i)11-s + (0.0108 − 0.0261i)13-s − 1.00·17-s + (0.975 + 0.404i)19-s + (−1.01 − 1.01i)23-s + (0.758 − 0.758i)25-s + (−0.539 + 1.30i)29-s − 1.77i·31-s + (0.169 + 0.0700i)35-s + (0.0288 + 0.0697i)37-s + (1.19 − 1.19i)41-s + (−0.456 − 1.10i)43-s − 0.891i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0619 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0619 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7196589328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7196589328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.97 - 1.23i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.237 + 0.237i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.12 + 0.879i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-0.0390 + 0.0943i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 + (-4.25 - 1.76i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (4.84 + 4.84i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.90 - 7.01i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 9.88iT - 31T^{2} \) |
| 37 | \( 1 + (-0.175 - 0.424i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-7.67 + 7.67i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.99 + 7.23i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 6.10iT - 47T^{2} \) |
| 53 | \( 1 + (4.28 + 10.3i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.19 - 2.88i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-6.29 - 2.60i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-5.66 + 13.6i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-3.49 + 3.49i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.42 + 1.42i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 + (2.95 - 7.14i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-1.92 - 1.92i)T + 89iT^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.562352331005430789867743183381, −8.648927127326529813628715657088, −7.913236252388197379275759720550, −7.11813545430393548602015862050, −6.44407850340375433024463535756, −5.27436583407286317021990315847, −3.97654426478449053102777830921, −3.65297669327926913416963725744, −2.23637367133706373004311621827, −0.34240183523661388658348542484,
1.28579348256318553622071391425, 2.93437774519598454685807574571, 4.06470026536977112571519111294, 4.57538265427541980622165603525, 5.77344024669472671587533847790, 6.82456022025814757768251602263, 7.65852250528276827335568014835, 8.265933238745930302850682040871, 9.223252861558371396976765297574, 9.769575883154950650176290895642