| L(s) = 1 | + (1.80 + 0.867i)2-s + (1.61 + 7.07i)3-s + (2.49 + 3.12i)4-s + (2.21 + 9.71i)5-s + (−3.22 + 14.1i)6-s + (7.24 − 17.0i)7-s + (1.78 + 7.79i)8-s + (−23.0 + 11.1i)9-s + (−4.43 + 19.4i)10-s + (−20.2 − 9.73i)11-s + (−18.0 + 22.6i)12-s + (−5.12 − 2.46i)13-s + (27.8 − 24.4i)14-s + (−65.1 + 31.3i)15-s + (−3.56 + 15.5i)16-s + (−5.55 + 6.96i)17-s + ⋯ |
| L(s) = 1 | + (0.637 + 0.306i)2-s + (0.310 + 1.36i)3-s + (0.311 + 0.390i)4-s + (0.198 + 0.868i)5-s + (−0.219 + 0.962i)6-s + (0.391 − 0.920i)7-s + (0.0786 + 0.344i)8-s + (−0.854 + 0.411i)9-s + (−0.140 + 0.614i)10-s + (−0.553 − 0.266i)11-s + (−0.435 + 0.545i)12-s + (−0.109 − 0.0526i)13-s + (0.531 − 0.466i)14-s + (−1.12 + 0.539i)15-s + (−0.0556 + 0.243i)16-s + (−0.0792 + 0.0994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.38998 + 2.05573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38998 + 2.05573i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.80 - 0.867i)T \) |
| 7 | \( 1 + (-7.24 + 17.0i)T \) |
| good | 3 | \( 1 + (-1.61 - 7.07i)T + (-24.3 + 11.7i)T^{2} \) |
| 5 | \( 1 + (-2.21 - 9.71i)T + (-112. + 54.2i)T^{2} \) |
| 11 | \( 1 + (20.2 + 9.73i)T + (829. + 1.04e3i)T^{2} \) |
| 13 | \( 1 + (5.12 + 2.46i)T + (1.36e3 + 1.71e3i)T^{2} \) |
| 17 | \( 1 + (5.55 - 6.96i)T + (-1.09e3 - 4.78e3i)T^{2} \) |
| 19 | \( 1 - 89.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (44.0 + 55.2i)T + (-2.70e3 + 1.18e4i)T^{2} \) |
| 29 | \( 1 + (95.0 - 119. i)T + (-5.42e3 - 2.37e4i)T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-209. + 262. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 + (68.8 + 301. i)T + (-6.20e4 + 2.99e4i)T^{2} \) |
| 43 | \( 1 + (102. - 447. i)T + (-7.16e4 - 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-38.7 - 18.6i)T + (6.47e4 + 8.11e4i)T^{2} \) |
| 53 | \( 1 + (249. + 312. i)T + (-3.31e4 + 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-51.9 + 227. i)T + (-1.85e5 - 8.91e4i)T^{2} \) |
| 61 | \( 1 + (-399. + 500. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 - 357.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-427. - 536. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (1.02e3 - 494. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + 841.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-240. + 115. i)T + (3.56e5 - 4.47e5i)T^{2} \) |
| 89 | \( 1 + (1.16e3 - 561. i)T + (4.39e5 - 5.51e5i)T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−14.34689499356547492172018648285, −13.01308900769050659852970864556, −11.30882127173104041878039073674, −10.58019715139533321987216759816, −9.703563548949816303175114572118, −8.119892858009700050014312468716, −6.88739634898073122335397584882, −5.29952037516754946089381181284, −4.12054388113136026911328607553, −2.98601726748682408085623296410,
1.35403069418339791167360285199, 2.62665151211451575325264665827, 4.89413836549418159727862153338, 5.99739487914054871770461587632, 7.47147491638004424990197476561, 8.478717540035549070023838477061, 9.770936703956923955891005779358, 11.62791239636689142446891922695, 12.18769050092826931867353112865, 13.16117379808506013752535132169
