| L(s) = 1 | + (−8.57e3 − 5.84e4i)3-s − 1.60e7i·5-s − 4.81e8·7-s + (−3.33e9 + 1.00e9i)9-s − 1.87e10i·11-s − 5.33e9·13-s + (−9.39e11 + 1.37e11i)15-s − 1.04e12i·17-s + 9.71e12·19-s + (4.12e12 + 2.81e13i)21-s − 4.89e13i·23-s − 1.63e14·25-s + (8.71e13 + 1.86e14i)27-s + 3.58e14i·29-s + 6.00e14·31-s + ⋯ |
| L(s) = 1 | + (−0.145 − 0.989i)3-s − 1.64i·5-s − 1.70·7-s + (−0.957 + 0.287i)9-s − 0.721i·11-s − 0.0387·13-s + (−1.62 + 0.239i)15-s − 0.517i·17-s + 1.58·19-s + (0.247 + 1.68i)21-s − 1.18i·23-s − 1.71·25-s + (0.423 + 0.906i)27-s + 0.851i·29-s + 0.732·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.5564384200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5564384200\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.57e3 + 5.84e4i)T \) |
| good | 5 | \( 1 + 1.60e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 + 4.81e8T + 7.97e16T^{2} \) |
| 11 | \( 1 + 1.87e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 + 5.33e9T + 1.90e22T^{2} \) |
| 17 | \( 1 + 1.04e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 9.71e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + 4.89e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 3.58e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 6.00e14T + 6.71e29T^{2} \) |
| 37 | \( 1 - 3.37e14T + 2.31e31T^{2} \) |
| 41 | \( 1 - 1.97e16iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 3.72e16T + 4.67e32T^{2} \) |
| 47 | \( 1 + 2.49e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 8.92e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + 4.77e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 3.54e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 4.54e17T + 3.32e36T^{2} \) |
| 71 | \( 1 + 2.82e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 1.99e17T + 1.84e37T^{2} \) |
| 79 | \( 1 + 7.41e18T + 8.96e37T^{2} \) |
| 83 | \( 1 - 1.89e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 2.85e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 7.49e19T + 5.43e39T^{2} \) |
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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
−13.64580111938929920419013265508, −12.86269679780647400425207015618, −11.89705012768310467683679298644, −9.559689784324011420008296298259, −8.347055102395369977443548083032, −6.60455662821497274726969631976, −5.25460550268767656208697548491, −3.07927662733170967617341033931, −1.06807768100337297160643349936, −0.20639712453170339146951207095,
2.83771222741481846066173821097, 3.67135898851407116861928935870, 5.89823528848132977632692296828, 7.12576831080789696075805827426, 9.635185250209953986673243881892, 10.24680869891327916244816560363, 11.75031222497622974310612355191, 13.71962553255339488948269825257, 15.14099164141361225592593905763, 15.90697705509393510320296609381
