| L(s) = 1 | + (1.87 + 0.684i)2-s + (6.85 − 2.49i)3-s + (3.06 + 2.57i)4-s + (−0.811 + 4.60i)5-s + 14.5·6-s + (−0.602 + 3.41i)7-s + (4.00 + 6.92i)8-s + (20.0 − 16.8i)9-s + (−4.67 + 8.09i)10-s + (−13.1 − 22.7i)11-s + (27.4 + 9.97i)12-s + (−28.7 − 24.1i)13-s + (−3.46 + 6.00i)14-s + (5.91 + 33.5i)15-s + (2.77 + 15.7i)16-s + (8.68 − 7.29i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (1.31 − 0.479i)3-s + (0.383 + 0.321i)4-s + (−0.0726 + 0.411i)5-s + 0.992·6-s + (−0.0325 + 0.184i)7-s + (0.176 + 0.306i)8-s + (0.742 − 0.622i)9-s + (−0.147 + 0.256i)10-s + (−0.359 − 0.622i)11-s + (0.659 + 0.239i)12-s + (−0.614 − 0.515i)13-s + (−0.0661 + 0.114i)14-s + (0.101 + 0.577i)15-s + (0.0434 + 0.246i)16-s + (0.123 − 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.87271 + 0.216355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.87271 + 0.216355i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 - 0.684i)T \) |
| 37 | \( 1 + (185. + 128. i)T \) |
| good | 3 | \( 1 + (-6.85 + 2.49i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (0.811 - 4.60i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (0.602 - 3.41i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (13.1 + 22.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (28.7 + 24.1i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-8.68 + 7.29i)T + (853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (2.89 - 1.05i)T + (5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (59.2 - 102. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (105. + 183. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (0.174 + 0.146i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-7.57 + 13.1i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-107. - 607. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (78.1 + 443. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-409. - 343. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-34.0 + 193. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-564. + 205. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 - 819.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-30.8 + 174. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-100. + 84.1i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-92.7 - 526. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-759. + 1.31e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.06706400889343017408674864792, −13.32494264639596848187553471097, −12.29723140573124498217825626644, −10.88280801920275137116490274878, −9.320043937375822254676061650457, −8.037180277316153924002677258107, −7.24280761826284778296337013875, −5.59985437920868564449942891716, −3.57747021815766480492899964197, −2.42347440345264998547066957039,
2.28289461952286478078846827693, 3.78988763302025448829550882141, 4.98366580723744215948913154110, 7.04511893252912371992425953836, 8.420852830693115734326036144692, 9.521004061781664581406101714243, 10.56548903550688782866758860913, 12.17813998237154266004164599810, 13.07376256733467279418404491302, 14.26066707510474499301309037053
