| L(s) = 1 | + (−11.4 − 23.8i)2-s + (−60.3 − 88.4i)3-s + (−275. + 345. i)4-s + (−115. + 124. i)5-s + (−1.41e3 + 2.44e3i)6-s + (812. − 469. i)7-s + (4.78e3 + 1.09e3i)8-s + (−1.79e3 + 4.56e3i)9-s + (4.30e3 + 1.32e3i)10-s + (1.34e4 + 1.68e4i)11-s + (4.71e4 + 3.53e3i)12-s + (1.79e4 − 5.53e3i)13-s + (−2.04e4 − 1.39e4i)14-s + (1.80e4 + 2.71e3i)15-s + (−3.69e3 − 1.62e4i)16-s + (−3.50e4 + 3.25e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.716 − 1.48i)2-s + (−0.744 − 1.09i)3-s + (−1.07 + 1.34i)4-s + (−0.185 + 0.199i)5-s + (−1.09 + 1.89i)6-s + (0.338 − 0.195i)7-s + (1.16 + 0.266i)8-s + (−0.273 + 0.695i)9-s + (0.430 + 0.132i)10-s + (0.916 + 1.14i)11-s + (2.27 + 0.170i)12-s + (0.628 − 0.193i)13-s + (−0.533 − 0.363i)14-s + (0.356 + 0.0537i)15-s + (−0.0564 − 0.247i)16-s + (−0.419 + 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.561379 - 0.424855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.561379 - 0.424855i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.44e6 - 3.09e6i)T \) |
| good | 2 | \( 1 + (11.4 + 23.8i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (60.3 + 88.4i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (115. - 124. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-812. + 469. i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.34e4 - 1.68e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-1.79e4 + 5.53e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (3.50e4 - 3.25e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (1.05e5 - 4.13e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (7.23e4 - 1.09e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-2.50e4 + 3.66e4i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-4.11e4 + 5.49e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-7.45e5 - 4.30e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-8.58e5 + 4.13e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-7.95e5 + 9.97e5i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-1.19e7 - 3.69e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (2.18e6 + 9.56e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (5.47e6 - 4.10e5i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-5.14e6 - 1.31e7i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-4.91e6 + 3.26e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (1.85e6 + 6.00e6i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (-2.37e7 - 4.10e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.97e7 + 2.70e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (1.61e7 + 2.36e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-1.08e8 - 1.36e8i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.23907581381774389998972126849, −12.40206430911919372670482369661, −11.56371315513462024024426749097, −10.69833108957424953126458717427, −9.291999052213175576178590796970, −7.79989261451305007195584868609, −6.39256785325124401048040041509, −4.02647547737326934377029814274, −1.98606559164094181398568266324, −1.03599754719010780621175324041,
0.48689041039529391357911848621, 4.23840507703958895984308323275, 5.56419911596913853381481190985, 6.58159550179655569735245839212, 8.383816308600966648713109189259, 9.164448226678058803258794826969, 10.59054129592954085473071492972, 11.69084906234584177054034680763, 13.85911719255995747444788130899, 14.97087712424408036473909256568
