| L(s) = 1 | + (4.36e3 + 7.55e3i)5-s + (−1.17e4 + 4.28e4i)7-s + (1.29e5 + 7.47e4i)11-s − 7.58e5i·13-s + (1.25e6 − 2.17e6i)17-s + (1.04e6 − 6.05e5i)19-s + (3.59e6 − 2.07e6i)23-s + (−1.36e7 + 2.36e7i)25-s + 1.21e7i·29-s + (−2.04e8 − 1.17e8i)31-s + (−3.75e8 + 9.79e7i)35-s + (−2.18e5 − 3.78e5i)37-s − 1.80e7·41-s − 1.24e9·43-s + (2.10e8 + 3.65e8i)47-s + ⋯ |
| L(s) = 1 | + (0.624 + 1.08i)5-s + (−0.265 + 0.964i)7-s + (0.242 + 0.140i)11-s − 0.566i·13-s + (0.214 − 0.370i)17-s + (0.0970 − 0.0560i)19-s + (0.116 − 0.0671i)23-s + (−0.279 + 0.483i)25-s + 0.110i·29-s + (−1.28 − 0.740i)31-s + (−1.20 + 0.315i)35-s + (−0.000518 − 0.000898i)37-s − 0.0243·41-s − 1.28·43-s + (0.134 + 0.232i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0496 + 0.998i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0496 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.6716085155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6716085155\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.17e4 - 4.28e4i)T \) |
| good | 5 | \( 1 + (-4.36e3 - 7.55e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.29e5 - 7.47e4i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.58e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-1.25e6 + 2.17e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.04e6 + 6.05e5i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.59e6 + 2.07e6i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.21e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (2.04e8 + 1.17e8i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.18e5 + 3.78e5i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.80e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.24e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-2.10e8 - 3.65e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (3.11e9 + 1.79e9i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-2.77e9 + 4.80e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (8.18e9 - 4.72e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-5.15e8 + 8.92e8i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.70e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.47e9 - 1.42e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (3.67e9 + 6.37e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 4.75e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.27e10 - 3.94e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.41e10iT - 7.15e21T^{2} \) |
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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
−9.836079907836675842841669637662, −9.099362620210003897009784893609, −7.87253682410876267924086696384, −6.77857155437671514495120971005, −6.00164726745138984391393357336, −5.08914283015132016612111406567, −3.46169913679965653273191985799, −2.66309002300285797178169007958, −1.73321834786887976564898483358, −0.11927699688374502139974013322,
1.06913277134389822435566877509, 1.75435954478761296218542307676, 3.37626971840133831904289714983, 4.39435572624803386675660696876, 5.34173240232703690242022990874, 6.42374545021007828979027677187, 7.43431733845010066965990728991, 8.605732814384971183179469821721, 9.377030670734909262266682574929, 10.21882596479953504957587822354
