L(s) = 1 | + (−1.99 + 3.44i)3-s + (−1.63 + 0.941i)5-s + (−5.14 − 4.74i)7-s + (−3.42 − 5.93i)9-s + (−3.93 + 6.82i)11-s − 11.4i·13-s − 7.49i·15-s + (1.44 − 2.51i)17-s + (−15.0 − 26.0i)19-s + (26.6 − 8.30i)21-s + (33.3 − 19.2i)23-s + (−10.7 + 18.5i)25-s − 8.56·27-s − 27.8i·29-s + (−19.4 − 11.2i)31-s + ⋯ |
L(s) = 1 | + (−0.663 + 1.14i)3-s + (−0.326 + 0.188i)5-s + (−0.735 − 0.677i)7-s + (−0.380 − 0.659i)9-s + (−0.358 + 0.620i)11-s − 0.883i·13-s − 0.499i·15-s + (0.0852 − 0.147i)17-s + (−0.790 − 1.36i)19-s + (1.26 − 0.395i)21-s + (1.45 − 0.838i)23-s + (−0.429 + 0.743i)25-s − 0.317·27-s − 0.961i·29-s + (−0.628 − 0.362i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.114099 - 0.160843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114099 - 0.160843i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (5.14 + 4.74i)T \) |
good | 3 | \( 1 + (1.99 - 3.44i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (1.63 - 0.941i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (3.93 - 6.82i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 11.4iT - 169T^{2} \) |
| 17 | \( 1 + (-1.44 + 2.51i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 + 26.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-33.3 + 19.2i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 27.8iT - 841T^{2} \) |
| 31 | \( 1 + (19.4 + 11.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (39.4 - 22.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 40.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (71.5 - 41.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (23.2 + 13.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (5.20 - 9.01i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (19.1 - 11.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (29.6 - 51.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 38.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (6.98 - 12.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (44.3 - 25.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 89.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + (52.6 + 91.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 55.3T + 9.40e3T^{2} \) |
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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
−11.37992147757783827915177516854, −10.69705755951090605903108395346, −10.02714378500329993109797904842, −9.093475797654389263243199354224, −7.59372295734976459190687509535, −6.57519167934701808881744041836, −5.19348863100775487264698279919, −4.32153419772836824612652165276, −3.06400413257715554281916241307, −0.11371656378329650363855742507,
1.73015094523711141262642636064, 3.50665329629585684070345654213, 5.34078261083037930674506359481, 6.27636621369410391739695933699, 7.09247419343209938405354492983, 8.280151012149565549724775771620, 9.241947472782703152476926287902, 10.61836547056759986979243414451, 11.64297139042004761848547345539, 12.37850684391577976835498539325