| L(s) = 1 | + (0.721 + 3.16i)2-s + (−124. − 115. i)3-s + (451. − 217. i)4-s + (2.33e3 − 352. i)5-s + (274. − 474. i)6-s + (3.67e3 + 6.36e3i)7-s + (2.04e3 + 2.56e3i)8-s + (668. + 8.92e3i)9-s + (2.80e3 + 7.13e3i)10-s + (2.77e4 + 1.33e4i)11-s + (−8.10e4 − 2.50e4i)12-s + (−5.45e4 + 1.39e5i)13-s + (−1.74e4 + 1.61e4i)14-s + (−3.30e5 − 2.25e5i)15-s + (1.53e5 − 1.92e5i)16-s + (3.66e5 + 5.53e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0318 + 0.139i)2-s + (−0.884 − 0.820i)3-s + (0.882 − 0.424i)4-s + (1.67 − 0.252i)5-s + (0.0863 − 0.149i)6-s + (0.578 + 1.00i)7-s + (0.176 + 0.221i)8-s + (0.0339 + 0.453i)9-s + (0.0885 + 0.225i)10-s + (0.571 + 0.275i)11-s + (−1.12 − 0.348i)12-s + (−0.529 + 1.35i)13-s + (−0.121 + 0.112i)14-s + (−1.68 − 1.15i)15-s + (0.585 − 0.734i)16-s + (1.06 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.67154 - 0.333446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.67154 - 0.333446i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.89e6 + 2.22e7i)T \) |
| good | 2 | \( 1 + (-0.721 - 3.16i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (124. + 115. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-2.33e3 + 352. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (-3.67e3 - 6.36e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.77e4 - 1.33e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (5.45e4 - 1.39e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-3.66e5 - 5.53e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (7.30e4 - 9.75e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-6.87e5 + 4.68e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-1.43e5 + 1.32e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (5.07e6 + 1.56e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (1.86e6 - 3.22e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (6.49e6 + 2.84e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-3.30e7 + 1.59e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (2.00e7 + 5.11e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (7.68e7 - 9.63e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (1.50e8 - 4.62e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (4.46e6 - 5.95e7i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.28e8 + 8.73e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (-3.05e7 + 7.79e7i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-5.32e7 - 9.21e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (3.19e8 + 2.96e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (1.68e8 + 1.56e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (1.23e9 + 5.94e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.08804836386868048265355982989, −12.33952764096496090202578203399, −11.90548513791337030828015872578, −10.37876597966577849319016240930, −9.140243277795010899868559132408, −7.09513955866771254536224356687, −5.97908536474888773342768012488, −5.46722913303950930287835252496, −1.95276315058281912222893691769, −1.53628883551050414570406809203,
1.19363234491065454682438824410, 2.94270642194868182296638654421, 4.94534050651633154446608018580, 6.07794026965169140362392481493, 7.44651804164704919111696836943, 9.633144120772413990639793235402, 10.63402160491466881884432118983, 11.15205673405612358493466433243, 12.80659080640207029225731117363, 14.00835447385238250798077748165
