| L(s) = 1 | + (80.7 + 58.6i)2-s + (−728. + 2.24e3i)3-s + (548. + 1.68e3i)4-s + (−1.29e4 + 9.38e3i)5-s + (−1.90e5 + 1.38e5i)6-s + (9.44e4 + 2.90e5i)7-s + (1.97e5 − 6.09e5i)8-s + (−3.21e6 − 2.33e6i)9-s − 1.59e6·10-s + (5.82e6 + 7.41e5i)11-s − 4.18e6·12-s + (−8.46e6 − 6.14e6i)13-s + (−9.42e6 + 2.90e7i)14-s + (−1.16e7 − 3.58e7i)15-s + (6.35e7 − 4.61e7i)16-s + (−1.20e8 + 8.73e7i)17-s + ⋯ |
| L(s) = 1 | + (0.892 + 0.648i)2-s + (−0.577 + 1.77i)3-s + (0.0669 + 0.205i)4-s + (−0.369 + 0.268i)5-s + (−1.66 + 1.21i)6-s + (0.303 + 0.933i)7-s + (0.267 − 0.821i)8-s + (−2.01 − 1.46i)9-s − 0.503·10-s + (0.992 + 0.126i)11-s − 0.404·12-s + (−0.486 − 0.353i)13-s + (−0.334 + 1.03i)14-s + (−0.263 − 0.811i)15-s + (0.946 − 0.687i)16-s + (−1.20 + 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.420294 - 1.56250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.420294 - 1.56250i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-5.82e6 - 7.41e5i)T \) |
| good | 2 | \( 1 + (-80.7 - 58.6i)T + (2.53e3 + 7.79e3i)T^{2} \) |
| 3 | \( 1 + (728. - 2.24e3i)T + (-1.28e6 - 9.37e5i)T^{2} \) |
| 5 | \( 1 + (1.29e4 - 9.38e3i)T + (3.77e8 - 1.16e9i)T^{2} \) |
| 7 | \( 1 + (-9.44e4 - 2.90e5i)T + (-7.83e10 + 5.69e10i)T^{2} \) |
| 13 | \( 1 + (8.46e6 + 6.14e6i)T + (9.35e13 + 2.88e14i)T^{2} \) |
| 17 | \( 1 + (1.20e8 - 8.73e7i)T + (3.06e15 - 9.41e15i)T^{2} \) |
| 19 | \( 1 + (2.64e7 - 8.12e7i)T + (-3.40e16 - 2.47e16i)T^{2} \) |
| 23 | \( 1 + 3.67e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-1.35e9 - 4.17e9i)T + (-8.30e18 + 6.03e18i)T^{2} \) |
| 31 | \( 1 + (-2.90e9 - 2.11e9i)T + (7.54e18 + 2.32e19i)T^{2} \) |
| 37 | \( 1 + (1.51e9 + 4.65e9i)T + (-1.97e20 + 1.43e20i)T^{2} \) |
| 41 | \( 1 + (-9.54e9 + 2.93e10i)T + (-7.48e20 - 5.43e20i)T^{2} \) |
| 43 | \( 1 + 2.41e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (3.45e10 - 1.06e11i)T + (-4.41e21 - 3.20e21i)T^{2} \) |
| 53 | \( 1 + (5.08e10 + 3.69e10i)T + (8.04e21 + 2.47e22i)T^{2} \) |
| 59 | \( 1 + (-1.46e11 - 4.49e11i)T + (-8.49e22 + 6.17e22i)T^{2} \) |
| 61 | \( 1 + (-4.95e10 + 3.59e10i)T + (5.00e22 - 1.53e23i)T^{2} \) |
| 67 | \( 1 + 8.27e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-1.52e12 + 1.10e12i)T + (3.60e23 - 1.10e24i)T^{2} \) |
| 73 | \( 1 + (-1.36e11 - 4.21e11i)T + (-1.35e24 + 9.82e23i)T^{2} \) |
| 79 | \( 1 + (-7.47e11 - 5.43e11i)T + (1.44e24 + 4.43e24i)T^{2} \) |
| 83 | \( 1 + (-4.19e12 + 3.04e12i)T + (2.74e24 - 8.43e24i)T^{2} \) |
| 89 | \( 1 - 3.21e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-1.06e13 - 7.75e12i)T + (2.07e25 + 6.40e25i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.48997701592951478305154030167, −16.07187798612494869255332466837, −15.18177624620031210867085580324, −14.54713351770718546237896893735, −12.09974496668603542277515441984, −10.62087113725591027968105447309, −9.102904288504929015984400147290, −6.18975078792270535732283176063, −4.92540538039856115839175045007, −3.75230081010075575852436298881,
0.57809334067833150810858814393, 2.17023055773188015653237774026, 4.50652588428969625499161364362, 6.62447786980391984849889318257, 8.044271900757009181202884749653, 11.38407185586203108425847459878, 11.98675495780852163232847038726, 13.35171494684074641344717793313, 14.04442652269223727286926580704, 16.90850601446990601529702128329
