| L(s) = 1 | + (−8.94 − 18.5i)2-s + (−59.7 − 87.5i)3-s + (−105. + 132. i)4-s + (802. − 864. i)5-s + (−1.09e3 + 1.89e3i)6-s + (−2.02e3 + 1.16e3i)7-s + (−1.74e3 − 397. i)8-s + (−1.70e3 + 4.35e3i)9-s + (−2.32e4 − 7.17e3i)10-s + (−8.25e3 − 1.03e4i)11-s + (1.78e4 + 1.34e3i)12-s + (−6.42e3 + 1.98e3i)13-s + (3.98e4 + 2.71e4i)14-s + (−1.23e5 − 1.86e4i)15-s + (1.78e4 + 7.82e4i)16-s + (−3.92e3 + 3.64e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.559 − 1.16i)2-s + (−0.737 − 1.08i)3-s + (−0.412 + 0.516i)4-s + (1.28 − 1.38i)5-s + (−0.843 + 1.46i)6-s + (−0.843 + 0.486i)7-s + (−0.425 − 0.0971i)8-s + (−0.260 + 0.663i)9-s + (−2.32 − 0.717i)10-s + (−0.563 − 0.706i)11-s + (0.863 + 0.0646i)12-s + (−0.224 + 0.0693i)13-s + (1.03 + 0.706i)14-s + (−2.44 − 0.368i)15-s + (0.272 + 1.19i)16-s + (−0.0469 + 0.0435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.583521 + 0.311982i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583521 + 0.311982i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.42e6 + 3.10e6i)T \) |
| good | 2 | \( 1 + (8.94 + 18.5i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (59.7 + 87.5i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (-802. + 864. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (2.02e3 - 1.16e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (8.25e3 + 1.03e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (6.42e3 - 1.98e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (3.92e3 - 3.64e3i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (2.75e4 - 1.07e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (-4.60e5 + 6.93e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-5.51e5 + 8.08e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (9.06e4 - 1.20e6i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (1.93e6 + 1.11e6i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-3.74e6 + 1.80e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (3.39e6 - 4.25e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (3.24e6 + 1.00e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-1.03e6 - 4.51e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-1.77e6 + 1.32e5i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (3.48e6 + 8.87e6i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-4.87e6 + 3.23e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-6.61e6 - 2.14e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (-2.11e7 - 3.67e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (4.55e7 - 3.10e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (2.94e7 + 4.32e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-2.46e7 - 3.09e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−12.59727615775546649893987825962, −12.41505012283254097166746690974, −10.83335606660859586934009552021, −9.546609224895988416759349678602, −8.685716926450657544151565912937, −6.39972190409641449007958513456, −5.44870878453424983733808098483, −2.53756473188996101542063093175, −1.26272985058137886449530377389, −0.35072223120518677760551364163,
2.94136520783219985728251495716, 5.22227715400669101797540300418, 6.38668648089341434891355046089, 7.21087404831711000920309577674, 9.456517662333890735700372457979, 10.07895285326645729362102832346, 11.02934347581495156922667340835, 13.14487009176761936792509199302, 14.60502478144831810264314625092, 15.33916566971721763437703565259
