| L(s) = 1 | + (−7.47 + 9.37i)2-s + (−35.4 + 5.33i)3-s + (−3.49 − 15.2i)4-s + (−37.7 + 25.7i)5-s + (214. − 371. i)6-s + (−223. − 387. i)7-s + (−1.21e3 − 584. i)8-s + (−864. + 266. i)9-s + (40.9 − 546. i)10-s + (1.76e3 − 7.74e3i)11-s + (205. + 523. i)12-s + (748. + 9.98e3i)13-s + (5.30e3 + 799. i)14-s + (1.20e3 − 1.11e3i)15-s + (1.63e4 − 7.87e3i)16-s + (1.41e4 + 9.65e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.660 + 0.828i)2-s + (−0.757 + 0.114i)3-s + (−0.0272 − 0.119i)4-s + (−0.135 + 0.0921i)5-s + (0.405 − 0.702i)6-s + (−0.246 − 0.427i)7-s + (−0.837 − 0.403i)8-s + (−0.395 + 0.121i)9-s + (0.0129 − 0.172i)10-s + (0.400 − 1.75i)11-s + (0.0343 + 0.0873i)12-s + (0.0944 + 1.26i)13-s + (0.516 + 0.0778i)14-s + (0.0918 − 0.0852i)15-s + (0.997 − 0.480i)16-s + (0.699 + 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.644904 + 0.100113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.644904 + 0.100113i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.23e5 + 5.06e5i)T \) |
| good | 2 | \( 1 + (7.47 - 9.37i)T + (-28.4 - 124. i)T^{2} \) |
| 3 | \( 1 + (35.4 - 5.33i)T + (2.08e3 - 644. i)T^{2} \) |
| 5 | \( 1 + (37.7 - 25.7i)T + (2.85e4 - 7.27e4i)T^{2} \) |
| 7 | \( 1 + (223. + 387. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.76e3 + 7.74e3i)T + (-1.75e7 - 8.45e6i)T^{2} \) |
| 13 | \( 1 + (-748. - 9.98e3i)T + (-6.20e7 + 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-1.41e4 - 9.65e3i)T + (1.49e8 + 3.81e8i)T^{2} \) |
| 19 | \( 1 + (1.53e4 + 4.74e3i)T + (7.38e8 + 5.03e8i)T^{2} \) |
| 23 | \( 1 + (-4.31e4 - 4.00e4i)T + (2.54e8 + 3.39e9i)T^{2} \) |
| 29 | \( 1 + (-1.69e5 - 2.56e4i)T + (1.64e10 + 5.08e9i)T^{2} \) |
| 31 | \( 1 + (9.02e4 + 2.30e5i)T + (-2.01e10 + 1.87e10i)T^{2} \) |
| 37 | \( 1 + (-2.64e5 + 4.58e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (4.37e5 - 5.48e5i)T + (-4.33e10 - 1.89e11i)T^{2} \) |
| 47 | \( 1 + (-9.85e4 - 4.31e5i)T + (-4.56e11 + 2.19e11i)T^{2} \) |
| 53 | \( 1 + (-9.96e4 + 1.32e6i)T + (-1.16e12 - 1.75e11i)T^{2} \) |
| 59 | \( 1 + (1.66e4 - 7.99e3i)T + (1.55e12 - 1.94e12i)T^{2} \) |
| 61 | \( 1 + (1.38e5 - 3.53e5i)T + (-2.30e12 - 2.13e12i)T^{2} \) |
| 67 | \( 1 + (-3.53e6 - 1.08e6i)T + (5.00e12 + 3.41e12i)T^{2} \) |
| 71 | \( 1 + (-1.55e6 + 1.44e6i)T + (6.79e11 - 9.06e12i)T^{2} \) |
| 73 | \( 1 + (4.17e5 + 5.57e6i)T + (-1.09e13 + 1.64e12i)T^{2} \) |
| 79 | \( 1 + (3.83e5 + 6.63e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.25e6 + 7.91e5i)T + (2.59e13 - 7.99e12i)T^{2} \) |
| 89 | \( 1 + (-6.58e6 + 9.92e5i)T + (4.22e13 - 1.30e13i)T^{2} \) |
| 97 | \( 1 + (-7.63e5 + 3.34e6i)T + (-7.27e13 - 3.50e13i)T^{2} \) |
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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
−14.74820774989498557995536640001, −13.47742319569032918477038916718, −11.80982235265368267036843645997, −10.99649658494682400670888631785, −9.293445483764881004787480856426, −8.196455526390547371257783818519, −6.72045644075050857997648095633, −5.77136041480408276769867020682, −3.55559659089993088077719868409, −0.51533382936559822834907663870,
0.931841158564404387207563627000, 2.72674887200978116132718496393, 5.12899790554303065418245935178, 6.53907568053308434270559516457, 8.443888002787696019259345198984, 9.819213700383308574474674027501, 10.66697704163871763652652209516, 12.12149838236959427253335433514, 12.38484183207484653723531567916, 14.56939485359679358167292800613
