| L(s) = 1 | + (−0.700 − 3.07i)5-s + (2.15 + 1.03i)7-s + (−2.67 − 2.13i)11-s + (2.48 − 3.11i)13-s + 0.723i·17-s + (−0.261 − 0.543i)19-s + (−0.402 + 1.76i)23-s + (−4.43 + 2.13i)25-s + (−0.849 − 5.31i)29-s + (−8.55 + 1.95i)31-s + (1.67 − 7.33i)35-s + (3.03 − 2.42i)37-s − 8.84i·41-s + (2.91 + 0.665i)43-s + (0.0107 + 0.00857i)47-s + ⋯ |
| L(s) = 1 | + (−0.313 − 1.37i)5-s + (0.813 + 0.391i)7-s + (−0.807 − 0.643i)11-s + (0.688 − 0.862i)13-s + 0.175i·17-s + (−0.0600 − 0.124i)19-s + (−0.0839 + 0.367i)23-s + (−0.886 + 0.427i)25-s + (−0.157 − 0.987i)29-s + (−1.53 + 0.350i)31-s + (0.283 − 1.24i)35-s + (0.499 − 0.398i)37-s − 1.38i·41-s + (0.444 + 0.101i)43-s + (0.00156 + 0.00125i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322085732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322085732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 29 | \( 1 + (0.849 + 5.31i)T \) |
| good | 5 | \( 1 + (0.700 + 3.07i)T + (-4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-2.15 - 1.03i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (2.67 + 2.13i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.48 + 3.11i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 - 0.723iT - 17T^{2} \) |
| 19 | \( 1 + (0.261 + 0.543i)T + (-11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (0.402 - 1.76i)T + (-20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (8.55 - 1.95i)T + (27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (-3.03 + 2.42i)T + (8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + 8.84iT - 41T^{2} \) |
| 43 | \( 1 + (-2.91 - 0.665i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.0107 - 0.00857i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.486 - 2.13i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 + (-0.0283 + 0.0587i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + (7.75 + 9.72i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (-3.03 + 3.81i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-9.14 - 2.08i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + (4.06 - 3.24i)T + (17.5 - 77.0i)T^{2} \) |
| 83 | \( 1 + (-9.47 + 4.56i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (16.2 - 3.70i)T + (80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (3.71 + 7.72i)T + (-60.4 + 75.8i)T^{2} \) |
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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
−9.440237282161122289197006343256, −8.679144646520549041965767226512, −8.153536917574066733379894232396, −7.50363129488817283907635513211, −5.84869076198673925357806208845, −5.41001504668179544797819237497, −4.50940808364082444777513496126, −3.42926622873658380696458762211, −1.93726476581089618858904990939, −0.59984859501761902538546090261,
1.73998633518971957806523066546, 2.88993840799842219251194791171, 3.95826868794926507103004921295, 4.86648482691729108744878683239, 6.06646136473538128512156195883, 7.01234530298534979499841875653, 7.51503368760776138148385175895, 8.365758057068817269519171975495, 9.461665802958360760355710953347, 10.36790857789039476483996626070
