| L(s) = 1 | − 1.41·2-s + (−2.79 + 1.09i)3-s + 2.00·4-s + (3.95 − 1.54i)6-s − 2.64i·7-s − 2.82·8-s + (6.62 − 6.09i)9-s − 3.30i·11-s + (−5.58 + 2.18i)12-s + 13.2i·13-s + 3.74i·14-s + 4.00·16-s − 14.2·17-s + (−9.36 + 8.61i)18-s − 7.45·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.931 + 0.363i)3-s + 0.500·4-s + (0.658 − 0.256i)6-s − 0.377i·7-s − 0.353·8-s + (0.735 − 0.677i)9-s − 0.300i·11-s + (−0.465 + 0.181i)12-s + 1.01i·13-s + 0.267i·14-s + 0.250·16-s − 0.836·17-s + (−0.520 + 0.478i)18-s − 0.392·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0916 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0916 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6161942693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6161942693\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (2.79 - 1.09i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 3.30iT - 121T^{2} \) |
| 13 | \( 1 - 13.2iT - 169T^{2} \) |
| 17 | \( 1 + 14.2T + 289T^{2} \) |
| 19 | \( 1 + 7.45T + 361T^{2} \) |
| 23 | \( 1 + 7.07T + 529T^{2} \) |
| 29 | \( 1 + 25.7iT - 841T^{2} \) |
| 31 | \( 1 - 41.5T + 961T^{2} \) |
| 37 | \( 1 + 27.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.12iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 54.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 62.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 21.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 84.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 54.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 116.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 23.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 3.26iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.5iT - 9.40e3T^{2} \) |
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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
−9.972613165110291257915501083456, −9.221126471341078846243996705275, −8.399052749530553484809555824803, −7.28657771071787519358166322060, −6.55974078056888421826284040856, −5.88125137442348648590220032070, −4.62075236694666953136697802838, −3.89552750954762231862243546103, −2.27422200100559485564652831496, −0.883603940927266036464404728654,
0.36467082825615263501569985325, 1.67028301205885239416934955057, 2.84335962209576489178768433578, 4.43524308918279603298235403075, 5.39618960763472455680998653243, 6.27462983604115353041474335125, 6.95551234044002099605056888339, 7.910174757122636280041859879444, 8.586820531263519516415556713905, 9.678006108876589605600933092381
