L(s) = 1 | + (63.7 − 5.98i)2-s + (652. − 652. i)3-s + (4.02e3 − 763. i)4-s + (−7.91e3 + 7.91e3i)5-s + (3.76e4 − 4.55e4i)6-s + 1.82e5·7-s + (2.51e5 − 7.27e4i)8-s − 3.20e5i·9-s + (−4.57e5 + 5.52e5i)10-s + (2.73e4 + 2.73e4i)11-s + (2.12e6 − 3.12e6i)12-s + (−5.51e6 − 5.51e6i)13-s + (1.16e7 − 1.09e6i)14-s + 1.03e7i·15-s + (1.56e7 − 6.14e6i)16-s − 9.08e6·17-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0935i)2-s + (0.895 − 0.895i)3-s + (0.982 − 0.186i)4-s + (−0.506 + 0.506i)5-s + (0.807 − 0.975i)6-s + 1.55·7-s + (0.960 − 0.277i)8-s − 0.603i·9-s + (−0.457 + 0.552i)10-s + (0.0154 + 0.0154i)11-s + (0.712 − 1.04i)12-s + (−1.14 − 1.14i)13-s + (1.54 − 0.145i)14-s + 0.907i·15-s + (0.930 − 0.366i)16-s − 0.376·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(4.23964 - 1.80078i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.23964 - 1.80078i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-63.7 + 5.98i)T \) |
good | 3 | \( 1 + (-652. + 652. i)T - 5.31e5iT^{2} \) |
| 5 | \( 1 + (7.91e3 - 7.91e3i)T - 2.44e8iT^{2} \) |
| 7 | \( 1 - 1.82e5T + 1.38e10T^{2} \) |
| 11 | \( 1 + (-2.73e4 - 2.73e4i)T + 3.13e12iT^{2} \) |
| 13 | \( 1 + (5.51e6 + 5.51e6i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 + 9.08e6T + 5.82e14T^{2} \) |
| 19 | \( 1 + (5.30e6 - 5.30e6i)T - 2.21e15iT^{2} \) |
| 23 | \( 1 - 8.24e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (5.25e8 + 5.25e8i)T + 3.53e17iT^{2} \) |
| 31 | \( 1 - 1.04e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + (3.53e9 - 3.53e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 - 2.81e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + (-5.67e9 - 5.67e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 + 1.79e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + (1.91e10 - 1.91e10i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + (3.17e10 + 3.17e10i)T + 1.77e21iT^{2} \) |
| 61 | \( 1 + (-1.08e10 - 1.08e10i)T + 2.65e21iT^{2} \) |
| 67 | \( 1 + (5.50e10 - 5.50e10i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 + 5.61e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + 9.77e10iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 3.19e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (-3.63e11 + 3.63e11i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 - 6.91e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 1.28e11T + 6.93e23T^{2} \) |
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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
−15.15906330033312244517122513942, −14.61093114029269593681845210613, −13.42030435135266383999390264131, −12.09612890241255782736251382080, −10.81104197518915509925551777029, −8.045566487620880248392494440394, −7.22558864503016042353207799206, −4.96674581421804452602013271810, −2.98683991734965526339651748878, −1.66130407142888369207326871183,
2.12331292762713194668371666897, 4.07433402216318529140384445441, 4.89666235785950243900573904333, 7.53315442861476885204686065391, 8.965337638678329899388941506373, 10.99056718350662808358591962745, 12.23492952278812048305530680464, 14.14896457273950221688798766799, 14.74157578027620566646363548196, 15.84499460719005120137434779296