L(s) = 1 | − 10.2·2-s + (−56.7 + 57.8i)3-s − 151.·4-s + (580. − 591. i)6-s + 3.44e3i·7-s + 4.16e3·8-s + (−122. − 6.55e3i)9-s + 7.12e3i·11-s + (8.58e3 − 8.74e3i)12-s − 4.13e4i·13-s − 3.52e4i·14-s − 3.95e3·16-s + 1.19e5·17-s + (1.25e3 + 6.71e4i)18-s + 8.62e4·19-s + ⋯ |
L(s) = 1 | − 0.639·2-s + (−0.700 + 0.713i)3-s − 0.590·4-s + (0.448 − 0.456i)6-s + 1.43i·7-s + 1.01·8-s + (−0.0187 − 0.999i)9-s + 0.486i·11-s + (0.413 − 0.421i)12-s − 1.44i·13-s − 0.918i·14-s − 0.0602·16-s + 1.43·17-s + (0.0119 + 0.639i)18-s + 0.662·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.480180 + 0.672838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.480180 + 0.672838i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (56.7 - 57.8i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 10.2T + 256T^{2} \) |
| 7 | \( 1 - 3.44e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 7.12e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.13e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.19e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 8.62e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.17e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 5.98e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.02e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 8.77e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.55e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.56e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 8.98e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 6.22e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 3.96e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 5.63e5T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.34e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.56e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.70e5iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.78e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 2.25e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 5.73e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 3.31e7iT - 7.83e15T^{2} \) |
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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
−12.87804504993388379438170782125, −12.08002283812356431821995861264, −10.72412017384345285379188723266, −9.746932191421959866390902926711, −8.973027826697457354404430451107, −7.69307416790668317115279698883, −5.69327390948119651166423258632, −5.01935988696801219285690970745, −3.21560083470266677574504873330, −0.963744591558344784446948117642,
0.53361105122599892292477641338, 1.38489228808479627488199734731, 3.93066479550509738478584809528, 5.30969523967388711024119985212, 7.01198443143739380072665131800, 7.69578225928250350504287322986, 9.160009529356933705364349888417, 10.38002954595419624992938942862, 11.23522853403879232903736140853, 12.56800378900069050737180581878