| L(s) = 1 | + 2.66i·5-s + 2.66i·7-s − 11-s − 3.60·13-s + 4.66i·17-s − 3.04i·19-s − 0.476·23-s − 2.12·25-s + 3.46i·29-s − 1.19i·31-s − 7.12·35-s − 8.37·37-s − 1.19i·41-s − 10.7i·43-s − 8.76·47-s + ⋯ |
| L(s) = 1 | + 1.19i·5-s + 1.00i·7-s − 0.301·11-s − 0.998·13-s + 1.13i·17-s − 0.698i·19-s − 0.0994·23-s − 0.424·25-s + 0.643i·29-s − 0.215i·31-s − 1.20·35-s − 1.37·37-s − 0.187i·41-s − 1.64i·43-s − 1.27·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8327147846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8327147846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.66iT - 5T^{2} \) |
| 7 | \( 1 - 2.66iT - 7T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 - 4.66iT - 17T^{2} \) |
| 19 | \( 1 + 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.476T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 1.19iT - 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 1.19iT - 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 4.25iT - 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 0.398T + 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 0.476T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−9.925654303575972190446820885231, −8.959914953539673294311295572747, −8.279433596252335188467073215268, −7.22676120546735586074428584847, −6.71646950793045687228462806429, −5.73237338963311115043588208244, −5.00900540701195168047012265853, −3.70433216604179662108671439911, −2.76443589278568024537783598836, −2.02747863766621956983319332685,
0.31130209652741792743516003725, 1.51918714491891845126502692868, 2.92323707866970612510288015959, 4.15377266795003702840518530865, 4.83314793868954069857104024795, 5.51641594925902291086847753610, 6.77063703082848247167356042272, 7.50300572898692324815807602693, 8.203374340210252651359241087888, 9.046718383827205889956766848057
