L(s) = 1 | + (−554. + 1.02e5i)3-s − 3.49e7i·5-s − 7.47e8i·7-s + (−1.04e10 − 1.13e8i)9-s − 1.00e11·11-s + 2.55e11·13-s + (3.57e12 + 1.93e10i)15-s − 6.80e12i·17-s − 4.43e13i·19-s + (7.64e13 + 4.14e11i)21-s − 2.67e14·23-s − 7.46e14·25-s + (1.73e13 − 1.06e15i)27-s − 4.19e15i·29-s + 4.81e15i·31-s + ⋯ |
L(s) = 1 | + (−0.00541 + 0.999i)3-s − 1.60i·5-s − 1.00i·7-s + (−0.999 − 0.0108i)9-s − 1.16·11-s + 0.513·13-s + (1.60 + 0.00868i)15-s − 0.818i·17-s − 1.65i·19-s + (1.00 + 0.00542i)21-s − 1.34·23-s − 1.56·25-s + (0.0162 − 0.999i)27-s − 1.85i·29-s + 1.05i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.6755443958\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6755443958\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (554. - 1.02e5i)T \) |
good | 5 | \( 1 + 3.49e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 + 7.47e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 1.00e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 2.55e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.80e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 + 4.43e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 2.67e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.19e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 - 4.81e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 3.54e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.43e17iT - 7.38e33T^{2} \) |
| 43 | \( 1 - 1.16e17iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 1.48e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.61e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 1.11e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.49e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.07e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 2.20e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.19e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.21e19iT - 7.08e39T^{2} \) |
| 83 | \( 1 - 2.26e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.74e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 - 3.33e20T + 5.27e41T^{2} \) |
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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
−10.68103540710416801440601583894, −9.650038361270890398374162466307, −8.698755641424293600366061511795, −7.65985509784001465732823082987, −5.75269124142206629209617864180, −4.72720890828247721793277020355, −4.13251044998445328809047676897, −2.56943567437666453674647241741, −0.802809754223126628346379819487, −0.16782351334002741993408480418,
1.72191973283182073088058812818, 2.52978296176736825462849018769, 3.48493487233734692512762833658, 5.74154297163145128129808314602, 6.25826959889963564393551769277, 7.59216146933868684114268373923, 8.340925008993857320308147135761, 10.11425998705941257600516754068, 11.07274855809189372070977708072, 12.13621481628190700247822685778