| L(s) = 1 | + 7.97i·2-s + (29.8 + 75.3i)3-s + 192.·4-s + (−600. + 238. i)6-s + 3.21e3·7-s + 3.57e3i·8-s + (−4.77e3 + 4.49e3i)9-s − 6.48e3i·11-s + (5.74e3 + 1.44e4i)12-s + 1.05e4·13-s + 2.56e4i·14-s + 2.07e4·16-s + 5.77e4i·17-s + (−3.58e4 − 3.81e4i)18-s + 2.28e5·19-s + ⋯ |
| L(s) = 1 | + 0.498i·2-s + (0.368 + 0.929i)3-s + 0.751·4-s + (−0.463 + 0.183i)6-s + 1.33·7-s + 0.873i·8-s + (−0.728 + 0.685i)9-s − 0.442i·11-s + (0.276 + 0.698i)12-s + 0.367·13-s + 0.666i·14-s + 0.316·16-s + 0.691i·17-s + (−0.341 − 0.363i)18-s + 1.75·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.85804 + 2.73497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85804 + 2.73497i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-29.8 - 75.3i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7.97iT - 256T^{2} \) |
| 7 | \( 1 - 3.21e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 6.48e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.05e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.77e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.28e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.13e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.11e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.04e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 6.30e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 4.62e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.44e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 6.69e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.25e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 5.55e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.31e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.70e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.08e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.67e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.07e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.32e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.21e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 9.03e7T + 7.83e15T^{2} \) |
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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
−13.67383352096497950360353531047, −11.58847216614354839295218363856, −11.18786858043336892568389424377, −9.877076883230887511350159242919, −8.372795096076230261282607226038, −7.71923614825388856031285257753, −5.92433608998151335145394874673, −4.85777667816309211210155163777, −3.23753120771725546213325174085, −1.66060702147421025963325544051,
1.06564925476390531264647565200, 1.94553829620516722549930024986, 3.28565540487221636040669744412, 5.30379610895401529721186087372, 6.95780434709252531639857559183, 7.68986376727890173127636906328, 9.023714130553604017427475204263, 10.60972496287434090285092124397, 11.68649139334548548654232492192, 12.21343490432880945648956173013
