| L(s) = 1 | + (26.6 + 36.6i)2-s + (−215. − 662. i)3-s + (−632. + 1.94e3i)4-s + (−6.73e3 − 4.88e3i)5-s + (1.85e4 − 2.54e4i)6-s + (1.27e5 + 4.14e4i)7-s + (−8.81e4 + 2.86e4i)8-s + (3.75e4 − 2.72e4i)9-s − 3.76e5i·10-s + (−1.74e6 + 2.99e5i)11-s + 1.42e6·12-s + (−3.66e6 − 5.03e6i)13-s + (1.87e6 + 5.76e6i)14-s + (−1.79e6 + 5.51e6i)15-s + (−3.39e6 − 2.46e6i)16-s + (−2.22e7 + 3.06e7i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.295 − 0.908i)3-s + (−0.154 + 0.475i)4-s + (−0.430 − 0.312i)5-s + (0.397 − 0.546i)6-s + (1.08 + 0.351i)7-s + (−0.336 + 0.109i)8-s + (0.0706 − 0.0513i)9-s − 0.376i·10-s + (−0.985 + 0.169i)11-s + 0.477·12-s + (−0.758 − 1.04i)13-s + (0.248 + 0.765i)14-s + (−0.157 + 0.483i)15-s + (−0.202 − 0.146i)16-s + (−0.921 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0299987 - 0.249243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0299987 - 0.249243i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-26.6 - 36.6i)T \) |
| 11 | \( 1 + (1.74e6 - 2.99e5i)T \) |
| good | 3 | \( 1 + (215. + 662. i)T + (-4.29e5 + 3.12e5i)T^{2} \) |
| 5 | \( 1 + (6.73e3 + 4.88e3i)T + (7.54e7 + 2.32e8i)T^{2} \) |
| 7 | \( 1 + (-1.27e5 - 4.14e4i)T + (1.11e10 + 8.13e9i)T^{2} \) |
| 13 | \( 1 + (3.66e6 + 5.03e6i)T + (-7.19e12 + 2.21e13i)T^{2} \) |
| 17 | \( 1 + (2.22e7 - 3.06e7i)T + (-1.80e14 - 5.54e14i)T^{2} \) |
| 19 | \( 1 + (6.98e7 - 2.26e7i)T + (1.79e15 - 1.30e15i)T^{2} \) |
| 23 | \( 1 - 7.00e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (4.18e8 + 1.36e8i)T + (2.86e17 + 2.07e17i)T^{2} \) |
| 31 | \( 1 + (6.96e8 - 5.06e8i)T + (2.43e17 - 7.49e17i)T^{2} \) |
| 37 | \( 1 + (3.07e8 - 9.46e8i)T + (-5.32e18 - 3.86e18i)T^{2} \) |
| 41 | \( 1 + (-8.21e9 + 2.66e9i)T + (1.82e19 - 1.32e19i)T^{2} \) |
| 43 | \( 1 + 2.31e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-3.00e9 - 9.25e9i)T + (-9.40e19 + 6.82e19i)T^{2} \) |
| 53 | \( 1 + (1.62e10 - 1.17e10i)T + (1.51e20 - 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-1.78e10 + 5.49e10i)T + (-1.43e21 - 1.04e21i)T^{2} \) |
| 61 | \( 1 + (2.93e10 - 4.04e10i)T + (-8.20e20 - 2.52e21i)T^{2} \) |
| 67 | \( 1 + 5.55e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-6.21e10 - 4.51e10i)T + (5.07e21 + 1.56e22i)T^{2} \) |
| 73 | \( 1 + (-3.46e10 - 1.12e10i)T + (1.85e22 + 1.34e22i)T^{2} \) |
| 79 | \( 1 + (1.91e11 + 2.64e11i)T + (-1.82e22 + 5.61e22i)T^{2} \) |
| 83 | \( 1 + (-2.22e11 + 3.06e11i)T + (-3.30e22 - 1.01e23i)T^{2} \) |
| 89 | \( 1 + 3.32e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (7.24e10 - 5.26e10i)T + (2.14e23 - 6.59e23i)T^{2} \) |
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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
−14.73545126755452283553350417096, −12.91330956298723200681030539424, −12.45919093223207520641765086189, −10.81659853054700554263124122021, −8.373567103378181406416518358206, −7.53037341247914220783407400975, −5.89798991153869137380568061471, −4.46610136752527754743884550922, −2.04292902647804332503375069262, −0.07626840740642007664936533243,
2.22841389889385031054889336502, 4.22934567734554247751284294742, 5.01743199784209421607402463268, 7.35450807502403691337769147739, 9.317775226578844531139052473091, 10.90247720948665435994881864732, 11.27838525985393666641114105063, 13.13581696834715465573460493156, 14.55971766954604131303926006942, 15.50481964756676447582829724144
