L(s) = 1 | + 2.43·2-s + 3.92·4-s + (0.613 − 1.06i)5-s + (2.20 + 1.46i)7-s + 4.68·8-s + (1.49 − 2.58i)10-s + (1.74 − 3.02i)11-s + (−2.87 + 2.17i)13-s + (5.35 + 3.57i)14-s + 3.55·16-s − 4.52·17-s + (−0.677 − 1.17i)19-s + (2.40 − 4.17i)20-s + (4.25 − 7.36i)22-s + 0.673·23-s + ⋯ |
L(s) = 1 | + 1.72·2-s + 1.96·4-s + (0.274 − 0.475i)5-s + (0.831 + 0.554i)7-s + 1.65·8-s + (0.472 − 0.818i)10-s + (0.526 − 0.911i)11-s + (−0.797 + 0.603i)13-s + (1.43 + 0.954i)14-s + 0.888·16-s − 1.09·17-s + (−0.155 − 0.269i)19-s + (0.538 − 0.933i)20-s + (0.906 − 1.56i)22-s + 0.140·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.47549 - 0.297296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.47549 - 0.297296i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.20 - 1.46i)T \) |
| 13 | \( 1 + (2.87 - 2.17i)T \) |
good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 + (-0.613 + 1.06i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.74 + 3.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + (0.677 + 1.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.673T + 23T^{2} \) |
| 29 | \( 1 + (2.64 + 4.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.99 - 8.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + (3.61 + 6.25i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.48 + 7.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.58 - 4.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.95 - 8.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 0.803T + 59T^{2} \) |
| 61 | \( 1 + (2.32 + 4.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.06 + 1.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.52 + 4.37i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.04 + 10.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.90 - 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + (3.59 - 6.22i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.69270894370580237922295991087, −9.161714425913537558691705472007, −8.630336632123181810574956725750, −7.26105470175440864692586060313, −6.44047155783803627986230157971, −5.51790556811271048649968036947, −4.85823763844532304911867294943, −4.08637294010338376219162271916, −2.80937062956901233652855834075, −1.77755593261791955218931379751,
1.90527659181437268732762395719, 2.85322456826963322955635879248, 4.19595992681010281472348263402, 4.65282183536960826801077236451, 5.64860327275893829648436999983, 6.71330017940389703399153355659, 7.18700195303634675964156779693, 8.292732687324248233383961140501, 9.721255421259712619404428913121, 10.53186240754998124013411046538