| L(s) = 1 | + (−2.55e4 − 2.55e4i)2-s + (−3.82e7 + 3.82e7i)3-s − 2.99e9i·4-s + 1.95e12·6-s + (1.22e13 + 1.22e13i)7-s + (−1.85e14 + 1.85e14i)8-s − 1.06e15i·9-s − 6.03e16·11-s + (1.14e17 + 1.14e17i)12-s + (−1.61e17 + 1.61e17i)13-s − 6.25e17i·14-s − 3.35e18·16-s + (−3.87e19 − 3.87e19i)17-s + (−2.73e19 + 2.73e19i)18-s + 2.45e20i·19-s + ⋯ |
| L(s) = 1 | + (−0.389 − 0.389i)2-s + (−0.888 + 0.888i)3-s − 0.696i·4-s + 0.691·6-s + (0.368 + 0.368i)7-s + (−0.660 + 0.660i)8-s − 0.577i·9-s − 1.31·11-s + (0.618 + 0.618i)12-s + (−0.243 + 0.243i)13-s − 0.287i·14-s − 0.181·16-s + (−0.795 − 0.795i)17-s + (−0.224 + 0.224i)18-s + 0.852i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.2028023262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2028023262\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (2.55e4 + 2.55e4i)T + 4.29e9iT^{2} \) |
| 3 | \( 1 + (3.82e7 - 3.82e7i)T - 1.85e15iT^{2} \) |
| 7 | \( 1 + (-1.22e13 - 1.22e13i)T + 1.10e27iT^{2} \) |
| 11 | \( 1 + 6.03e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (1.61e17 - 1.61e17i)T - 4.42e35iT^{2} \) |
| 17 | \( 1 + (3.87e19 + 3.87e19i)T + 2.36e39iT^{2} \) |
| 19 | \( 1 - 2.45e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (-9.22e19 + 9.22e19i)T - 3.76e43iT^{2} \) |
| 29 | \( 1 + 1.82e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + 3.83e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-1.31e25 - 1.31e25i)T + 1.52e50iT^{2} \) |
| 41 | \( 1 + 1.00e26T + 4.06e51T^{2} \) |
| 43 | \( 1 + (7.50e25 - 7.50e25i)T - 1.86e52iT^{2} \) |
| 47 | \( 1 + (-3.79e24 - 3.79e24i)T + 3.21e53iT^{2} \) |
| 53 | \( 1 + (5.28e27 - 5.28e27i)T - 1.50e55iT^{2} \) |
| 59 | \( 1 + 2.33e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 6.44e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (1.20e29 + 1.20e29i)T + 2.71e58iT^{2} \) |
| 71 | \( 1 + 7.66e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-5.26e28 + 5.26e28i)T - 4.22e59iT^{2} \) |
| 79 | \( 1 - 2.15e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (4.04e30 - 4.04e30i)T - 2.57e61iT^{2} \) |
| 89 | \( 1 - 2.02e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-5.39e31 - 5.39e31i)T + 3.77e63iT^{2} \) |
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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
−11.07647835056088999588519103646, −10.26267808873343872076913309645, −9.429189313929858827456216962556, −8.023533078381024261767491911168, −6.22657245206059478862170733054, −5.22925701630268535963971832191, −4.62221753876292636202214136873, −2.76219307377451149306417573596, −1.66040481558214632588560587596, −0.14838767385922345221667301227,
0.30711190069723947899582667689, 1.73380524284456296785124843241, 3.06545883864479038386091370305, 4.62926484367353412259023064493, 5.91702998927006344576699934304, 7.04028854582934395958061029795, 7.69946172826336018897868895519, 8.848587500641963615042324338333, 10.55228475208151532882684635873, 11.56946995951923444330754193078
