L(s) = 1 | + (4.60e4 − 4.65e4i)2-s − 8.00e7i·3-s + (−4.59e7 − 4.29e9i)4-s − 3.25e10·5-s + (−3.73e12 − 3.69e12i)6-s + 4.74e13i·7-s + (−2.02e14 − 1.95e14i)8-s − 4.55e15·9-s + (−1.49e15 + 1.51e15i)10-s + 1.35e16i·11-s + (−3.43e17 + 3.67e15i)12-s + 9.45e16·13-s + (2.21e18 + 2.18e18i)14-s + 2.60e18i·15-s + (−1.84e19 + 3.94e17i)16-s + 6.35e18·17-s + ⋯ |
L(s) = 1 | + (0.703 − 0.710i)2-s − 1.85i·3-s + (−0.0106 − 0.999i)4-s − 0.213·5-s + (−1.32 − 1.30i)6-s + 1.42i·7-s + (−0.718 − 0.695i)8-s − 2.45·9-s + (−0.149 + 0.151i)10-s + 0.294i·11-s + (−1.85 + 0.0198i)12-s + 0.142·13-s + (1.01 + 1.00i)14-s + 0.396i·15-s + (−0.999 + 0.0213i)16-s + 0.130·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(1.126043129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126043129\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.60e4 + 4.65e4i)T \) |
good | 3 | \( 1 + 8.00e7iT - 1.85e15T^{2} \) |
| 5 | \( 1 + 3.25e10T + 2.32e22T^{2} \) |
| 7 | \( 1 - 4.74e13iT - 1.10e27T^{2} \) |
| 11 | \( 1 - 1.35e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 - 9.45e16T + 4.42e35T^{2} \) |
| 17 | \( 1 - 6.35e18T + 2.36e39T^{2} \) |
| 19 | \( 1 + 4.93e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + 4.92e21iT - 3.76e43T^{2} \) |
| 29 | \( 1 + 2.04e23T + 6.26e46T^{2} \) |
| 31 | \( 1 - 8.15e23iT - 5.29e47T^{2} \) |
| 37 | \( 1 - 8.64e24T + 1.52e50T^{2} \) |
| 41 | \( 1 + 9.37e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + 9.13e25iT - 1.86e52T^{2} \) |
| 47 | \( 1 + 4.69e26iT - 3.21e53T^{2} \) |
| 53 | \( 1 - 3.40e27T + 1.50e55T^{2} \) |
| 59 | \( 1 - 2.87e27iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 7.72e27T + 1.35e57T^{2} \) |
| 67 | \( 1 + 8.28e28iT - 2.71e58T^{2} \) |
| 71 | \( 1 - 2.37e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + 4.61e27T + 4.22e59T^{2} \) |
| 79 | \( 1 + 3.06e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 - 3.67e29iT - 2.57e61T^{2} \) |
| 89 | \( 1 + 2.12e31T + 2.40e62T^{2} \) |
| 97 | \( 1 - 2.45e30T + 3.77e63T^{2} \) |
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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
−15.09684352795566957125222549284, −13.50480760027237797194152525133, −12.39612133969084622235574072687, −11.53009699565604306206571639801, −8.762066383304908450583588227775, −6.78684903504018693892475651472, −5.45413145572690612875919848553, −2.75936959394890153380692633726, −1.82302346163115416272555654209, −0.27698848101185851346731347330,
3.56118118734662589005560453815, 4.16085331745688948256740230658, 5.70550035631374908790399072473, 7.941219183433904717347664172650, 9.845163616208985422195770901012, 11.32727507847606811427552799316, 13.80178996427195333607106270202, 14.99408865330907150469509636271, 16.31701016894955683181246854640, 17.02632053762628560939976985435