| L(s) = 1 | + (−1.91 − 8.37i)2-s + (7.67 + 7.12i)3-s + (−37.7 + 18.1i)4-s + (59.1 − 8.91i)5-s + (44.9 − 77.9i)6-s + (−90.2 − 156. i)7-s + (52.9 + 66.3i)8-s + (−9.96 − 133. i)9-s + (−187. − 478. i)10-s + (115. + 55.7i)11-s + (−418. − 129. i)12-s + (−51.2 + 130. i)13-s + (−1.13e3 + 1.05e3i)14-s + (517. + 352. i)15-s + (−380. + 477. i)16-s + (106. + 16.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.338 − 1.48i)2-s + (0.492 + 0.456i)3-s + (−1.17 + 0.567i)4-s + (1.05 − 0.159i)5-s + (0.510 − 0.883i)6-s + (−0.696 − 1.20i)7-s + (0.292 + 0.366i)8-s + (−0.0410 − 0.547i)9-s + (−0.594 − 1.51i)10-s + (0.288 + 0.139i)11-s + (−0.839 − 0.259i)12-s + (−0.0841 + 0.214i)13-s + (−1.55 + 1.43i)14-s + (0.593 + 0.404i)15-s + (−0.371 + 0.465i)16-s + (0.0892 + 0.0134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.385227 - 1.50193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.385227 - 1.50193i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.05e3 - 9.86e3i)T \) |
| good | 2 | \( 1 + (1.91 + 8.37i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-7.67 - 7.12i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (-59.1 + 8.91i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (90.2 + 156. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-115. - 55.7i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (51.2 - 130. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (-106. - 16.0i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (-174. + 2.32e3i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (710. - 484. i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (418. - 387. i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (-9.89e3 - 3.05e3i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (2.66e3 - 4.60e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (508. + 2.22e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-2.05e4 + 9.90e3i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (-6.66e3 - 1.69e4i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (980. - 1.22e3i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-7.44e3 + 2.29e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (4.98e3 - 6.64e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (3.29e4 + 2.24e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (-1.55e4 + 3.95e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (-1.85e4 - 3.21e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-6.88e4 - 6.38e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (9.72e4 + 9.02e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-1.12e5 - 5.42e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−13.92287320360519544085131646335, −13.27429035046997576605553488609, −11.94937320953094556760541022640, −10.47444918024196605966747808260, −9.759714053305374606056880242411, −8.997856384824040311407215594947, −6.64020032608482708202823629087, −4.17943842733044770379791577786, −2.80969097676647414269680893841, −0.931719720592381702268455316445,
2.39080233291958865020421926835, 5.58345026905178979951502281113, 6.32994906809224430470450183405, 7.85571388884239631956422413097, 8.905808773968509412042050180517, 9.993243150645860429124473806015, 12.25128037678592098819720780061, 13.62647743636252813356861309638, 14.35302411375736114360425305728, 15.50430638643094822566686379484
