| L(s) = 1 | + (0.528 + 2.31i)2-s + (−14.7 − 13.6i)3-s + (23.7 − 11.4i)4-s + (−88.0 + 13.2i)5-s + (23.9 − 41.4i)6-s + (116. + 201. i)7-s + (86.3 + 108. i)8-s + (12.1 + 162. i)9-s + (−77.2 − 196. i)10-s + (298. + 143. i)11-s + (−507. − 156. i)12-s + (−291. + 742. i)13-s + (−405. + 375. i)14-s + (1.48e3 + 1.01e3i)15-s + (320. − 402. i)16-s + (−1.89e3 − 285. i)17-s + ⋯ |
| L(s) = 1 | + (0.0933 + 0.409i)2-s + (−0.947 − 0.878i)3-s + (0.742 − 0.357i)4-s + (−1.57 + 0.237i)5-s + (0.271 − 0.469i)6-s + (0.898 + 1.55i)7-s + (0.477 + 0.598i)8-s + (0.0499 + 0.667i)9-s + (−0.244 − 0.622i)10-s + (0.744 + 0.358i)11-s + (−1.01 − 0.313i)12-s + (−0.478 + 1.21i)13-s + (−0.552 + 0.512i)14-s + (1.70 + 1.15i)15-s + (0.313 − 0.393i)16-s + (−1.58 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.593690 + 0.686239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593690 + 0.686239i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.17e4 - 3.15e3i)T \) |
| good | 2 | \( 1 + (-0.528 - 2.31i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (14.7 + 13.6i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (88.0 - 13.2i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (-116. - 201. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-298. - 143. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (291. - 742. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (1.89e3 + 285. i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (79.0 - 1.05e3i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (-1.09e3 + 747. i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (2.73e3 - 2.53e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (-3.47e3 - 1.07e3i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (1.62e3 - 2.81e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.00e3 + 4.39e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (126. - 60.7i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (394. + 1.00e3i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (-4.61e3 + 5.78e3i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-3.23e4 + 9.97e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (1.54e3 - 2.05e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (-6.75e3 - 4.60e3i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (300. - 765. i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (5.17e4 + 8.96e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.88e4 + 1.74e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (-6.63e4 - 6.15e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-7.74e4 - 3.72e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−15.23172804993269407891662564929, −14.59556083237682128616687668051, −12.31341860429988027214167095607, −11.59779939853665815146243043860, −11.32923733056324678876724123676, −8.661960621268116000630139959514, −7.24822244096302990817342891036, −6.43074945030375121699052695247, −4.83046212393387358017185204410, −1.89699511979780353261545916273,
0.53595847433446883023802824624, 3.81111620646241021516135422977, 4.62365630241771679186232455177, 7.01098598773906909906972709997, 8.090001661722299943799514519655, 10.43092338608532514782951334945, 11.24086028860366815814046289007, 11.59368379467452762550400589491, 13.16228884708601273621525579933, 15.12160559516090960768727041332
