| L(s) = 1 | + (1.05 − 1.70i)2-s + (−1.93 + 2.29i)3-s + (−1.78 − 3.57i)4-s + (−0.00768 + 4.99i)5-s + (1.87 + 5.70i)6-s + (2.87 − 2.87i)7-s + (−7.96 − 0.718i)8-s + (−1.53 − 8.86i)9-s + (8.49 + 5.27i)10-s + (−6.98 − 5.07i)11-s + (11.6 + 2.80i)12-s + (−3.42 + 21.6i)13-s + (−1.86 − 7.91i)14-s + (−11.4 − 9.67i)15-s + (−9.60 + 12.7i)16-s + (2.01 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (0.525 − 0.850i)2-s + (−0.644 + 0.764i)3-s + (−0.447 − 0.894i)4-s + (−0.00153 + 0.999i)5-s + (0.312 + 0.950i)6-s + (0.410 − 0.410i)7-s + (−0.995 − 0.0898i)8-s + (−0.170 − 0.985i)9-s + (0.849 + 0.527i)10-s + (−0.634 − 0.461i)11-s + (0.972 + 0.233i)12-s + (−0.263 + 1.66i)13-s + (−0.133 − 0.565i)14-s + (−0.763 − 0.645i)15-s + (−0.600 + 0.799i)16-s + (0.118 + 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.454899 + 0.593822i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.454899 + 0.593822i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.05 + 1.70i)T \) |
| 3 | \( 1 + (1.93 - 2.29i)T \) |
| 5 | \( 1 + (0.00768 - 4.99i)T \) |
| good | 7 | \( 1 + (-2.87 + 2.87i)T - 49iT^{2} \) |
| 11 | \( 1 + (6.98 + 5.07i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.42 - 21.6i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-2.01 - 3.95i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (10.4 - 32.2i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (40.8 - 6.46i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (5.79 + 17.8i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (3.56 + 1.15i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-36.3 - 5.74i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (4.48 + 6.17i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-7.64 - 7.64i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.87 + 9.55i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-31.5 + 62.0i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (15.2 + 21.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (50.7 + 36.9i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-16.7 - 32.9i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (19.6 + 60.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-0.445 + 0.0705i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-36.7 - 113. i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-40.8 - 80.2i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-34.5 - 25.1i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-74.4 + 146. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.60914191342255473051857537122, −10.94289836481425458549449274588, −10.17422015005434907693421957072, −9.569332397759239555532479432829, −8.035070877552751930113612317209, −6.44558424016279270773308263842, −5.74759570491024727950344161989, −4.31452431353446459627111730550, −3.68758161560804355761284334285, −2.02568114099614058465532385684,
0.31534369901651818103828406618, 2.49579328131699857454395732171, 4.58563579480908379288452899651, 5.29801716999539784107371134796, 6.06358068361795963350546968860, 7.46879239730392016436912125471, 8.013805466916092358850955436526, 8.957327957428560874251830902066, 10.40359672901055812111328282288, 11.69848747576930500287912597319
