| L(s) = 1 | + (−1.82 + 2.29i)2-s + (−7.90 − 9.90i)3-s + (5.21 + 22.8i)4-s + (−7.91 − 3.81i)5-s + 37.1·6-s + 25.4·7-s + (−146. − 70.4i)8-s + (18.3 − 80.3i)9-s + (23.1 − 11.1i)10-s + (132. − 580. i)11-s + (185. − 232. i)12-s + (−615. − 296. i)13-s + (−46.4 + 58.2i)14-s + (24.7 + 108. i)15-s + (−246. + 118. i)16-s + (2.06e3 − 993. i)17-s + ⋯ |
| L(s) = 1 | + (−0.322 + 0.404i)2-s + (−0.506 − 0.635i)3-s + (0.162 + 0.713i)4-s + (−0.141 − 0.0682i)5-s + 0.420·6-s + 0.196·7-s + (−0.807 − 0.389i)8-s + (0.0754 − 0.330i)9-s + (0.0733 − 0.0353i)10-s + (0.329 − 1.44i)11-s + (0.371 − 0.465i)12-s + (−1.01 − 0.486i)13-s + (−0.0632 + 0.0793i)14-s + (0.0284 + 0.124i)15-s + (−0.241 + 0.116i)16-s + (1.73 − 0.833i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.618053 - 0.548443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618053 - 0.548443i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-391. + 1.21e4i)T \) |
| good | 2 | \( 1 + (1.82 - 2.29i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (7.90 + 9.90i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (7.91 + 3.81i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 - 25.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-132. + 580. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (615. + 296. i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-2.06e3 + 993. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (310. + 1.36e3i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (586. - 2.57e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (2.29e3 - 2.88e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (1.05e3 - 1.31e3i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 - 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (6.77e3 - 8.50e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (2.41e3 + 1.05e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.88e4 + 9.08e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (2.82e3 - 1.36e3i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-3.71e3 - 4.65e3i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (8.02e3 + 3.51e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-9.26e3 - 4.05e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (2.76e4 + 1.33e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + 3.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (2.16e4 + 2.72e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-7.69e4 - 9.64e4i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (4.31e3 - 1.89e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−14.80265527674123704236347321688, −13.33918166605257588124698777474, −12.12530902358350532157954308000, −11.50804360527418662820034016832, −9.525320231646819917335443883606, −8.074887326457929130410167500219, −7.08906888979302412110055067042, −5.69671528935964049335495049200, −3.28709038971411833345727011198, −0.52940943344395235786629682177,
1.86742036897195665083863375147, 4.46184211983672054279824823761, 5.82509822581773295417336082219, 7.65027411542205497023363424896, 9.733222837363799299316713396026, 10.14125779242298074499923113964, 11.46614915336767600642748407448, 12.45290113777853475062711552985, 14.54297483744488240196036340427, 14.99683489184190688750804452068
