| L(s) = 1 | + (−0.291 − 1.27i)2-s + (9.04 + 8.39i)3-s + (27.2 − 13.1i)4-s + (−0.113 + 0.0170i)5-s + (8.08 − 14.0i)6-s + (53.8 + 93.3i)7-s + (−50.9 − 63.8i)8-s + (−6.78 − 90.5i)9-s + (0.0548 + 0.139i)10-s + (410. + 197. i)11-s + (356. + 110. i)12-s + (−126. + 322. i)13-s + (103. − 96.1i)14-s + (−1.16 − 0.795i)15-s + (537. − 673. i)16-s + (1.89e3 + 285. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0515 − 0.225i)2-s + (0.580 + 0.538i)3-s + (0.852 − 0.410i)4-s + (−0.00202 + 0.000305i)5-s + (0.0917 − 0.158i)6-s + (0.415 + 0.719i)7-s + (−0.281 − 0.352i)8-s + (−0.0279 − 0.372i)9-s + (0.000173 + 0.000441i)10-s + (1.02 + 0.492i)11-s + (0.715 + 0.220i)12-s + (−0.207 + 0.529i)13-s + (0.141 − 0.131i)14-s + (−0.00133 − 0.000912i)15-s + (0.524 − 0.658i)16-s + (1.59 + 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0417i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.34357 + 0.0489011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34357 + 0.0489011i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.93e3 + 1.19e4i)T \) |
| good | 2 | \( 1 + (0.291 + 1.27i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-9.04 - 8.39i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (0.113 - 0.0170i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (-53.8 - 93.3i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-410. - 197. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (126. - 322. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (-1.89e3 - 285. i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (-157. + 2.09e3i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (3.20e3 - 2.18e3i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (4.41e3 - 4.09e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (5.73e3 + 1.76e3i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (428. - 742. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (2.11e3 + 9.24e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (1.15e4 - 5.54e3i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (-3.79e3 - 9.67e3i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (1.00e4 - 1.25e4i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (9.70e3 - 2.99e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (-1.41e3 + 1.88e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (-3.28e4 - 2.24e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (2.57e4 - 6.55e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (1.99e4 + 3.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.12e4 + 1.04e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (4.09e4 + 3.79e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-5.04e4 - 2.43e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−14.95252769242908889054068086435, −14.25259190882225356165181708623, −12.19280019016465796938803527598, −11.51064336118687302360898582126, −9.886004373787064615118880648825, −9.086701401849073932964374362211, −7.27854076946904928798784430553, −5.69303010805902910452507117794, −3.60988950276801330668642045656, −1.82336663585442903834464967202,
1.69789902562673290707235452971, 3.54580739374727719958911160708, 6.00594754469143451641554621243, 7.62043245989161635270647078603, 8.086796511869618969793464057066, 10.11595726441216179183772930319, 11.49688183751572785561650880451, 12.55062683503904624299206123810, 14.04381634792324583356927568803, 14.64529391400639750423497093982
