| L(s) = 1 | + (−18.0 − 22.5i)2-s + (152. + 23.0i)3-s + (−71.5 + 313. i)4-s + (827. + 564. i)5-s + (−2.22e3 − 3.86e3i)6-s + (−595. + 1.03e3i)7-s + (−4.95e3 + 2.38e3i)8-s + (3.99e3 + 1.23e3i)9-s + (−2.16e3 − 2.88e4i)10-s + (−6.84e3 − 2.99e4i)11-s + (−1.81e4 + 4.62e4i)12-s + (3.66e3 − 4.89e4i)13-s + (3.40e4 − 5.12e3i)14-s + (1.13e5 + 1.05e5i)15-s + (2.91e5 + 1.40e5i)16-s + (3.85e5 − 2.62e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.795 − 0.997i)2-s + (1.08 + 0.164i)3-s + (−0.139 + 0.612i)4-s + (0.591 + 0.403i)5-s + (−0.702 − 1.21i)6-s + (−0.0937 + 0.162i)7-s + (−0.427 + 0.206i)8-s + (0.202 + 0.0625i)9-s + (−0.0683 − 0.911i)10-s + (−0.140 − 0.617i)11-s + (−0.252 + 0.643i)12-s + (0.0355 − 0.475i)13-s + (0.236 − 0.0356i)14-s + (0.578 + 0.536i)15-s + (1.11 + 0.535i)16-s + (1.11 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.08855 - 1.50098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08855 - 1.50098i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.17e7 - 5.47e6i)T \) |
| good | 2 | \( 1 + (18.0 + 22.5i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (-152. - 23.0i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-827. - 564. i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (595. - 1.03e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (6.84e3 + 2.99e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (-3.66e3 + 4.89e4i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (-3.85e5 + 2.62e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-6.04e5 + 1.86e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (-7.41e5 + 6.88e5i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-2.02e6 + 3.05e5i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-1.01e6 + 2.59e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (7.06e6 + 1.22e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.33e5 - 1.67e5i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (2.12e6 - 9.29e6i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-1.92e6 - 2.56e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (1.07e8 + 5.15e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (1.18e6 + 3.02e6i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (-3.48e7 + 1.07e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-2.29e7 - 2.13e7i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (3.58e6 - 4.77e7i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-1.78e7 + 3.08e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.73e8 + 2.61e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (-2.19e8 - 3.31e7i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-3.19e8 - 1.39e9i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−13.77960586668844154465521458832, −12.21022912858887287322012821476, −10.90930667102773619704551221078, −9.806086129308126316998019539257, −9.053568892879366191205414155140, −7.84616586194163765166030279286, −5.76217456452984977780376940026, −3.18435167967698330166977981531, −2.51637466112857214369204393388, −0.796772022791273944155126322682,
1.43541379389997218041451988827, 3.27414906787826707423886967354, 5.55202010536648384572321368285, 7.13608714045024146933188593681, 8.110637167229522236448504453077, 9.136282608063601200420939994865, 9.971206371995328775715494165904, 12.19218217442011253059535287284, 13.52132764657637227756472849628, 14.49883903979625148906095145420
