| L(s) = 1 | + (−5.17 − 0.452i)2-s + (4.38 − 2.78i)3-s + (18.6 + 3.29i)4-s + (2.29 − 10.9i)5-s + (−23.9 + 12.4i)6-s + (−15.0 − 10.5i)7-s + (−55.0 − 14.7i)8-s + (11.4 − 24.4i)9-s + (−16.8 + 55.5i)10-s + (19.0 − 52.2i)11-s + (91.1 − 37.5i)12-s + (7.10 + 81.2i)13-s + (72.9 + 61.2i)14-s + (−20.4 − 54.3i)15-s + (135. + 49.2i)16-s + (−37.4 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.82 − 0.160i)2-s + (0.844 − 0.536i)3-s + (2.33 + 0.411i)4-s + (0.205 − 0.978i)5-s + (−1.62 + 0.845i)6-s + (−0.811 − 0.567i)7-s + (−2.43 − 0.651i)8-s + (0.425 − 0.905i)9-s + (−0.532 + 1.75i)10-s + (0.521 − 1.43i)11-s + (2.19 − 0.904i)12-s + (0.151 + 1.73i)13-s + (1.39 + 1.16i)14-s + (−0.351 − 0.936i)15-s + (2.11 + 0.769i)16-s + (−0.534 + 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.171959 - 0.716139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171959 - 0.716139i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.38 + 2.78i)T \) |
| 5 | \( 1 + (-2.29 + 10.9i)T \) |
| good | 2 | \( 1 + (5.17 + 0.452i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (15.0 + 10.5i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-19.0 + 52.2i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-7.10 - 81.2i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (37.4 - 10.0i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-85.1 + 49.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (44.0 + 62.8i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (192. - 161. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (11.1 - 63.1i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (23.2 + 86.6i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (9.82 - 11.7i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (111. + 239. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (29.9 - 42.8i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-73.9 + 73.9i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-57.0 + 20.7i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (64.8 + 367. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-60.1 + 5.26i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-44.9 - 25.9i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (2.27 - 8.48i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (-538. - 642. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-15.3 + 175. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-225. - 390. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.14e3 + 535. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.06866049588158467638656644151, −11.13476993788180424209314000388, −9.678799857532203589477365532724, −9.038507891306872832066025120516, −8.544461018072582282096087035818, −7.19703024000910522091950869770, −6.39600509361466191093082813218, −3.57652054350828187351259010606, −1.80994117895724856661385581452, −0.59003046505957088820481498223,
2.08658877947828042035166733355, 3.24346185694123050814492098584, 5.94530178674208274862050245920, 7.28467818493395929207706128857, 7.930778777627774502108417219350, 9.369363626307201870640733121487, 9.786639737387435917864266964623, 10.48295768558431103180143610847, 11.72748199843832749534684812997, 13.19404720595337630267101760668
