L(s) = 1 | + (0.707 − 0.707i)5-s + 1.47i·7-s + (1.91 − 1.91i)11-s + (4.06 + 4.06i)13-s + 1.32·17-s + (−3.46 − 3.46i)19-s − 7.30i·23-s − 1.00i·25-s + (4.04 + 4.04i)29-s − 0.0828·31-s + (1.04 + 1.04i)35-s + (4.52 − 4.52i)37-s + 0.696i·41-s + (−6.70 + 6.70i)43-s + 8.68·47-s + ⋯ |
L(s) = 1 | + (0.316 − 0.316i)5-s + 0.558i·7-s + (0.578 − 0.578i)11-s + (1.12 + 1.12i)13-s + 0.321·17-s + (−0.795 − 0.795i)19-s − 1.52i·23-s − 0.200i·25-s + (0.750 + 0.750i)29-s − 0.0148·31-s + (0.176 + 0.176i)35-s + (0.744 − 0.744i)37-s + 0.108i·41-s + (−1.02 + 1.02i)43-s + 1.26·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160921608\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160921608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - 1.47iT - 7T^{2} \) |
| 11 | \( 1 + (-1.91 + 1.91i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.06 - 4.06i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + (3.46 + 3.46i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.30iT - 23T^{2} \) |
| 29 | \( 1 + (-4.04 - 4.04i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.0828T + 31T^{2} \) |
| 37 | \( 1 + (-4.52 + 4.52i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.696iT - 41T^{2} \) |
| 43 | \( 1 + (6.70 - 6.70i)T - 43iT^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + (4.80 - 4.80i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.484 + 0.484i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.60 + 6.60i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.600 + 0.600i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.46iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 2.25T + 79T^{2} \) |
| 83 | \( 1 + (-8.28 - 8.28i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.83iT - 89T^{2} \) |
| 97 | \( 1 - 6.90T + 97T^{2} \) |
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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
−8.727122903662363418083217446103, −8.383844086370444746705228274873, −7.08100419445938805025459977266, −6.32300115346325849471192007897, −5.93492940270811212870355565454, −4.74771349639322673652146277328, −4.14948059055222091673052442357, −3.02651582965263212617659409783, −2.03391557823222371510118630066, −0.918267237974787942408459955921,
0.984499415232483023536527117579, 2.00140433711815093598232997909, 3.32123480011176329085919923224, 3.86043235693964495898115803188, 4.90737770439470747770394650765, 5.93832099230021875223549724217, 6.33957802631200940659480669385, 7.39948970874068343922893415187, 7.919095554257848247414370376654, 8.780535249371929314690688298999