L(s) = 1 | + (−0.300 + 0.826i)2-s + (−1.62 + 0.592i)3-s + (0.939 + 0.788i)4-s − 1.52i·6-s + (2.41 + 2.87i)7-s + (−2.45 + 1.41i)8-s + (2.29 − 1.92i)9-s + (−0.180 − 1.02i)11-s + (−1.99 − 0.726i)12-s + (1.08 + 2.99i)13-s + (−3.10 + 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (0.405 + 0.233i)17-s + (0.902 + 2.47i)18-s + (2.34 + 4.06i)19-s + ⋯ |
L(s) = 1 | + (−0.212 + 0.584i)2-s + (−0.939 + 0.342i)3-s + (0.469 + 0.394i)4-s − 0.621i·6-s + (0.913 + 1.08i)7-s + (−0.868 + 0.501i)8-s + (0.766 − 0.642i)9-s + (−0.0545 − 0.309i)11-s + (−0.576 − 0.209i)12-s + (0.302 + 0.830i)13-s + (−0.830 + 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (0.0982 + 0.0567i)17-s + (0.212 + 0.584i)18-s + (0.538 + 0.932i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.299477 + 1.11371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.299477 + 1.11371i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.62 - 0.592i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.300 - 0.826i)T + (-1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 2.87i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.180 + 1.02i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.08 - 2.99i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.405 - 0.233i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.45 + 4.11i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.45 - 1.98i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.14 + 2.63i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.87 + 2.23i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.52 - 2.73i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (11.9 - 2.11i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.22 - 2.65i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 8.83iT - 53T^{2} \) |
| 59 | \( 1 + (2.36 - 13.4i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.46 + 6.26i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.623 - 1.71i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.85 - 6.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.705 - 0.407i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.81 + 1.38i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.81 + 15.9i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.19 + 9.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.02 + 1.06i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.05312524606941857985269184006, −10.05199772338805068061851271331, −8.835807097141086630356295885230, −8.376879326129695064316019531152, −7.19822586576491121654454155627, −6.36906141086876484103325878677, −5.59569017339344113577429643267, −4.75978404052805324008822641485, −3.33252427444800281087361549331, −1.79262514607155529390763129522,
0.77684381566177173291517508519, 1.71934796596379884188946755897, 3.32583475421166310092946574270, 4.81376765721897112364693820760, 5.48152341916769402909594854920, 6.78661549703887560042213391658, 7.23736173727124023100736265727, 8.321329773263874770060750416802, 9.718952594818606986551115182898, 10.45219764938990611391045663651