| L(s) = 1 | + (−2.76 + 0.591i)2-s + (−5.08 + 1.09i)3-s + (7.30 − 3.27i)4-s + (−6.87 + 18.8i)5-s + (13.4 − 6.02i)6-s + (22.1 + 3.89i)7-s + (−18.2 + 13.3i)8-s + (24.6 − 11.0i)9-s + (7.85 − 56.3i)10-s + (−8.65 + 3.15i)11-s + (−33.5 + 24.5i)12-s + (−50.5 + 42.4i)13-s + (−63.4 + 2.29i)14-s + (14.3 − 103. i)15-s + (42.6 − 47.7i)16-s + (−28.8 + 16.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.977 + 0.209i)2-s + (−0.977 + 0.210i)3-s + (0.912 − 0.408i)4-s + (−0.615 + 1.69i)5-s + (0.912 − 0.409i)6-s + (1.19 + 0.210i)7-s + (−0.806 + 0.590i)8-s + (0.911 − 0.410i)9-s + (0.248 − 1.78i)10-s + (−0.237 + 0.0863i)11-s + (−0.806 + 0.591i)12-s + (−1.07 + 0.905i)13-s + (−1.21 + 0.0437i)14-s + (0.246 − 1.78i)15-s + (0.665 − 0.746i)16-s + (−0.411 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0662461 - 0.273877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0662461 - 0.273877i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.76 - 0.591i)T \) |
| 3 | \( 1 + (5.08 - 1.09i)T \) |
| good | 5 | \( 1 + (6.87 - 18.8i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-22.1 - 3.89i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (8.65 - 3.15i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (50.5 - 42.4i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (28.8 - 16.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (47.1 + 27.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (24.3 + 138. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-60.0 + 71.5i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (13.8 - 2.43i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-171. - 296. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-39.0 - 46.5i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (123. + 340. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (38.5 - 218. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 71.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (695. + 253. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-77.6 + 440. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-306. - 365. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-59.7 - 103. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (564. - 977. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (118. - 141. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (289. + 242. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-605. - 349. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-32.5 + 11.8i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−14.40920637095238158956130159773, −12.09844138332559471775465855323, −11.36777497305357870266319514083, −10.76879675881650167065787840410, −9.878380610901774331039278096744, −8.185927731539996958283069246989, −7.09563617409984670807365798613, −6.36878555028205528489616915150, −4.60670063619092007901860202226, −2.30571041907734462096893778750,
0.24214533849960001675615048181, 1.52744630017399874504318701623, 4.49987623719378878409782013254, 5.54116962109748663299896837956, 7.53162567530529222742861198588, 8.057204818780187011888903275998, 9.338511699078272773534021311226, 10.62969949818953100276538464003, 11.59469346088071161243105714240, 12.28760080538679769279059736712
