| L(s) = 1 | + (−3.71 − 4.66i)2-s + (263. + 39.7i)3-s + (106. − 464. i)4-s + (1.63e3 + 1.11e3i)5-s + (−794. − 1.37e3i)6-s + (3.93e3 − 6.81e3i)7-s + (−5.31e3 + 2.55e3i)8-s + (4.90e4 + 1.51e4i)9-s + (−883. − 1.17e4i)10-s + (−7.62e3 − 3.34e4i)11-s + (4.63e4 − 1.18e5i)12-s + (−1.31e3 + 1.75e4i)13-s + (−4.63e4 + 6.99e3i)14-s + (3.86e5 + 3.58e5i)15-s + (−1.88e5 − 9.05e4i)16-s + (−3.21e5 + 2.19e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.164 − 0.206i)2-s + (1.87 + 0.282i)3-s + (0.207 − 0.907i)4-s + (1.17 + 0.798i)5-s + (−0.250 − 0.433i)6-s + (0.618 − 1.07i)7-s + (−0.458 + 0.220i)8-s + (2.48 + 0.767i)9-s + (−0.0279 − 0.372i)10-s + (−0.157 − 0.688i)11-s + (0.645 − 1.64i)12-s + (−0.0127 + 0.170i)13-s + (−0.322 + 0.0486i)14-s + (1.97 + 1.83i)15-s + (−0.717 − 0.345i)16-s + (−0.934 + 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.16313 - 1.12330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.16313 - 1.12330i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-9.91e6 + 2.01e7i)T \) |
| good | 2 | \( 1 + (3.71 + 4.66i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (-263. - 39.7i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-1.63e3 - 1.11e3i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-3.93e3 + 6.81e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (7.62e3 + 3.34e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (1.31e3 - 1.75e4i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (3.21e5 - 2.19e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (5.11e5 - 1.57e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (1.84e6 - 1.71e6i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-2.09e6 + 3.16e5i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-2.43e6 + 6.20e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (1.62e6 + 2.81e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.88e6 - 3.61e6i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (5.82e6 - 2.55e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-8.26e6 - 1.10e8i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-4.19e6 - 2.01e6i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (2.55e7 + 6.50e7i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (-6.26e7 + 1.93e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (9.05e7 + 8.39e7i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (-1.72e6 + 2.29e7i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-2.30e8 + 3.98e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.22e7 - 4.85e6i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (5.54e8 + 8.36e7i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.12e8 - 4.93e8i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−13.87868125624866745107758063421, −13.58617035883784308939613417705, −10.78655091223283606782849192582, −10.17835223903276307517873407345, −9.162546932076881179396768640837, −7.78646352678954456706104999639, −6.26663135254216203038554285540, −4.10955385704488388764946598267, −2.44282280329922013326698291505, −1.60790664837758549700202921771,
1.98407418705853817391059686089, 2.54339020672659332258241420814, 4.53647543182465587179875724433, 6.74780238914231570600145901112, 8.392563753454006832009957604880, 8.654277960694845811036727591631, 9.794853204870172213801041418607, 12.30719526119061781492431628041, 12.98449755412016392114004640194, 14.01736158772507037503824915706
