| L(s) = 1 | + (0.573 + 0.819i)2-s + (−0.342 + 0.939i)4-s + (0.718 + 0.0628i)5-s + (−2.07 + 1.73i)7-s + (−0.965 + 0.258i)8-s + (0.360 + 0.624i)10-s + (−2.14 + 3.71i)11-s + (0.0350 − 0.0163i)13-s + (−2.61 − 0.699i)14-s + (−0.766 − 0.642i)16-s + (1.26 + 0.588i)17-s + (−1.15 − 0.812i)19-s + (−0.304 + 0.653i)20-s + (−4.27 + 0.374i)22-s + (−1.21 + 4.54i)23-s + ⋯ |
| L(s) = 1 | + (0.405 + 0.579i)2-s + (−0.171 + 0.469i)4-s + (0.321 + 0.0280i)5-s + (−0.782 + 0.656i)7-s + (−0.341 + 0.0915i)8-s + (0.113 + 0.197i)10-s + (−0.647 + 1.12i)11-s + (0.00973 − 0.00453i)13-s + (−0.697 − 0.186i)14-s + (−0.191 − 0.160i)16-s + (0.306 + 0.142i)17-s + (−0.266 − 0.186i)19-s + (−0.0681 + 0.146i)20-s + (−0.911 + 0.0797i)22-s + (−0.253 + 0.947i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.379355 + 1.25910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379355 + 1.25910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.573 - 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-6.02 + 0.856i)T \) |
| good | 5 | \( 1 + (-0.718 - 0.0628i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (2.07 - 1.73i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (2.14 - 3.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0350 + 0.0163i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-1.26 - 0.588i)T + (10.9 + 13.0i)T^{2} \) |
| 19 | \( 1 + (1.15 + 0.812i)T + (6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (1.21 - 4.54i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.18 - 8.15i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (3.19 + 3.19i)T + 31iT^{2} \) |
| 41 | \( 1 + (-8.42 - 3.06i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.673 - 0.673i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.04 - 1.18i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.385 + 0.459i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.273 + 3.12i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (4.73 + 10.1i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-5.48 - 6.53i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.759 - 0.133i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 9.36iT - 73T^{2} \) |
| 79 | \( 1 + (0.124 - 1.41i)T + (-77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (-2.84 - 7.81i)T + (-63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 0.934i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (2.87 + 0.769i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.83578243249831495543331696342, −9.703503496841020575271671847097, −9.323558892969312692197464848309, −8.058219689095728063669719051624, −7.30802583102290765606355210880, −6.29034338394485242944685253639, −5.57716268526914867916106216980, −4.58945339959195572791038465765, −3.32773859822444630355365477982, −2.16434923857068970880821172225,
0.59127386009171844857832905370, 2.40072311555419776259020701676, 3.44229936301495829432917165727, 4.40181976366022348176718995287, 5.72352761421925329871075471532, 6.28087909796783515221999853079, 7.54272677911415530319705309578, 8.529212320816110626513967328066, 9.596855950860622108359529558232, 10.25815780684032392923860347175
