L(s) = 1 | + (3.45 − 15.6i)2-s + (−40.5 − 23.3i)3-s + (−232. − 108. i)4-s + (−48.5 − 84.0i)5-s + (−505. + 551. i)6-s + (−518. − 2.34e3i)7-s + (−2.49e3 + 3.25e3i)8-s + (1.09e3 + 1.89e3i)9-s + (−1.48e3 + 467. i)10-s + (−4.04e3 − 2.33e3i)11-s + (6.87e3 + 9.80e3i)12-s − 4.61e4·13-s + (−3.84e4 − 0.265i)14-s + 4.53e3i·15-s + (4.21e4 + 5.01e4i)16-s + (−2.66e4 + 4.62e4i)17-s + ⋯ |
L(s) = 1 | + (0.216 − 0.976i)2-s + (−0.5 − 0.288i)3-s + (−0.906 − 0.422i)4-s + (−0.0776 − 0.134i)5-s + (−0.389 + 0.425i)6-s + (−0.216 − 0.976i)7-s + (−0.608 + 0.793i)8-s + (0.166 + 0.288i)9-s + (−0.148 + 0.0467i)10-s + (−0.276 − 0.159i)11-s + (0.331 + 0.472i)12-s − 1.61·13-s + (−0.999 − 6.91e−6i)14-s + 0.0896i·15-s + (0.643 + 0.765i)16-s + (−0.319 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.383994 + 0.0467615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.383994 + 0.0467615i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.45 + 15.6i)T \) |
| 3 | \( 1 + (40.5 + 23.3i)T \) |
| 7 | \( 1 + (518. + 2.34e3i)T \) |
good | 5 | \( 1 + (48.5 + 84.0i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (4.04e3 + 2.33e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.61e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (2.66e4 - 4.62e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-7.21e4 + 4.16e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.79e5 - 1.61e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.26e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-9.68e5 - 5.58e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-2.65e4 - 4.59e4i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 2.70e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 6.37e4iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.86e6 + 4.54e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.08e5 + 3.61e5i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-5.34e6 - 3.08e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.20e6 + 1.07e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-3.09e7 - 1.78e7i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.44e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.85e7 - 3.22e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.86e7 - 1.07e7i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.37e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.98e7 - 3.44e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.59e8T + 7.83e15T^{2} \) |
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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
−12.47199287659009737825613923170, −11.75116081457088275903459415904, −10.45886900767273885484278672579, −9.878333260844989434154865273418, −8.228206784419465618680025176364, −6.84578255076874166141909996027, −5.23954211204876584216621931551, −4.16101217870042579451990308164, −2.55168678444585744391938737118, −0.960072138129073927475303814705,
0.14447153706551918191519453233, 2.77535237427775966333660129260, 4.57801665174460552772225843167, 5.49298167065428902904566879984, 6.67652872926067495382740364170, 7.85048091476430626516279584997, 9.216972002799344800418453533817, 10.09147703943409201320968815247, 11.90478507502726437995385617091, 12.47164879888116093168689898889