| L(s) = 1 | + (−0.614 + 2.29i)2-s + (−3.60 + 3.74i)3-s + (2.04 + 1.18i)4-s + (−2.27 − 2.27i)5-s + (−6.36 − 10.5i)6-s + (−25.9 + 6.94i)7-s + (−17.3 + 17.3i)8-s + (−1.02 − 26.9i)9-s + (6.62 − 3.82i)10-s + (59.0 + 15.8i)11-s + (−11.8 + 3.40i)12-s + (19.7 + 42.5i)13-s − 63.6i·14-s + (16.7 − 0.318i)15-s + (−19.7 − 34.1i)16-s + (3.91 − 6.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.217 + 0.810i)2-s + (−0.693 + 0.720i)3-s + (0.256 + 0.147i)4-s + (−0.203 − 0.203i)5-s + (−0.433 − 0.718i)6-s + (−1.39 + 0.374i)7-s + (−0.768 + 0.768i)8-s + (−0.0379 − 0.999i)9-s + (0.209 − 0.121i)10-s + (1.61 + 0.433i)11-s + (−0.284 + 0.0819i)12-s + (0.420 + 0.907i)13-s − 1.21i·14-s + (0.288 − 0.00547i)15-s + (−0.308 − 0.534i)16-s + (0.0558 − 0.0967i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.146990 + 0.812315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146990 + 0.812315i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.60 - 3.74i)T \) |
| 13 | \( 1 + (-19.7 - 42.5i)T \) |
| good | 2 | \( 1 + (0.614 - 2.29i)T + (-6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (2.27 + 2.27i)T + 125iT^{2} \) |
| 7 | \( 1 + (25.9 - 6.94i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-59.0 - 15.8i)T + (1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (-3.91 + 6.78i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-29.4 - 110. i)T + (-5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-5.49 - 9.51i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-21.6 + 12.4i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-34.2 + 34.2i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-35.8 + 133. i)T + (-4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (53.8 - 200. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (76.2 + 44.0i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-273. + 273. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + 309. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-144. - 540. i)T + (-1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-73.2 + 126. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (179. + 47.9i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (660. - 176. i)T + (3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-512. - 512. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 866.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-440. - 440. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (437. + 117. i)T + (6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (193. + 721. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−16.51093052336621677071512248642, −15.49969134201445494527150225048, −14.38151627267400978375362712114, −12.28604779940102545866758210931, −11.68691269076672699340689565190, −9.828448205497094643258676167518, −8.856429961011676438411546685171, −6.77323705679384178059968988073, −6.02118972369162301286126011966, −3.82158398475915718631462357657,
0.838434026256610484941626899371, 3.26521363674178592897462705947, 6.15697862310288467252624964256, 7.00132741808817294369514835790, 9.260478414696569199528173739418, 10.60462577378165525002323661829, 11.53921418124757633235174830033, 12.56879397129354731327073925290, 13.59799489338760697523819027765, 15.42961969579863737498914395161
