| L(s) = 1 | + (−1.76 − 2.20i)2-s + (4.72 − 2.16i)3-s + (−1.76 + 7.80i)4-s + (−1.80 + 4.96i)5-s + (−13.1 − 6.61i)6-s + (9.40 + 1.65i)7-s + (20.3 − 9.88i)8-s + (17.6 − 20.4i)9-s + (14.1 − 4.77i)10-s + (53.5 − 19.4i)11-s + (8.56 + 40.6i)12-s + (−18.5 + 15.5i)13-s + (−12.9 − 23.7i)14-s + (2.21 + 27.3i)15-s + (−57.7 − 27.5i)16-s + (35.7 − 20.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.624 − 0.781i)2-s + (0.909 − 0.416i)3-s + (−0.220 + 0.975i)4-s + (−0.161 + 0.443i)5-s + (−0.893 − 0.449i)6-s + (0.507 + 0.0895i)7-s + (0.899 − 0.436i)8-s + (0.652 − 0.757i)9-s + (0.447 − 0.150i)10-s + (1.46 − 0.534i)11-s + (0.206 + 0.978i)12-s + (−0.396 + 0.332i)13-s + (−0.247 − 0.452i)14-s + (0.0380 + 0.470i)15-s + (−0.902 − 0.430i)16-s + (0.509 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42228 - 0.857795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42228 - 0.857795i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.76 + 2.20i)T \) |
| 3 | \( 1 + (-4.72 + 2.16i)T \) |
| good | 5 | \( 1 + (1.80 - 4.96i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-9.40 - 1.65i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-53.5 + 19.4i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (18.5 - 15.5i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-35.7 + 20.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-21.9 - 12.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.1 + 148. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-40.0 + 47.7i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (258. - 45.5i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (14.6 + 25.3i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-261. - 311. i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-59.7 - 164. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-1.92 + 10.8i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 528. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (545. + 198. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-28.2 + 159. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-577. - 687. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-82.7 - 143. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (121. - 210. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-120. + 143. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (963. + 808. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (704. + 406. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (23.1 - 8.41i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.83315082901902096419947645148, −11.92811927920299465558052900229, −11.00435517425311104888411198023, −9.616475340807434355527892905601, −8.832684796241274571846883246881, −7.77547864896653944539781552590, −6.71448786336321683382467593247, −4.15591041324881680770187270913, −2.89973672706635820299928045403, −1.34049119881139512460679681290,
1.56434872156905779562125822332, 4.02144788844254923194984414847, 5.29028598716098543583649395191, 7.07541164637182504233586636326, 7.996337113475276864930278682936, 9.065774605269142495294398071542, 9.714986092003088811212375933288, 10.99166470005989597770266979802, 12.46737771256433001297129947671, 13.92488818574529702485773112046
