L(s) = 1 | − 3.75e6i·2-s + (1.02e10 − 2.19e9i)3-s − 9.68e12·4-s + 5.14e14i·5-s + (−8.24e15 − 3.83e16i)6-s + 7.24e17·7-s + 1.98e19i·8-s + (9.97e19 − 4.49e19i)9-s + 1.93e21·10-s − 5.80e21i·11-s + (−9.90e22 + 2.12e22i)12-s − 2.70e23·13-s − 2.71e24i·14-s + (1.13e24 + 5.25e24i)15-s + 3.19e25·16-s − 9.64e25i·17-s + ⋯ |
L(s) = 1 | − 1.78i·2-s + (0.977 − 0.210i)3-s − 2.20·4-s + 1.07i·5-s + (−0.375 − 1.74i)6-s + 1.29·7-s + 2.15i·8-s + (0.911 − 0.410i)9-s + 1.93·10-s − 0.784i·11-s + (−2.15 + 0.462i)12-s − 1.09·13-s − 2.32i·14-s + (0.226 + 1.05i)15-s + 1.65·16-s − 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{43}{2})\) |
\(\approx\) |
\(2.695663742\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.695663742\) |
\(L(22)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.02e10 + 2.19e9i)T \) |
good | 2 | \( 1 + 3.75e6iT - 4.39e12T^{2} \) |
| 5 | \( 1 - 5.14e14iT - 2.27e29T^{2} \) |
| 7 | \( 1 - 7.24e17T + 3.11e35T^{2} \) |
| 11 | \( 1 + 5.80e21iT - 5.47e43T^{2} \) |
| 13 | \( 1 + 2.70e23T + 6.10e46T^{2} \) |
| 17 | \( 1 + 9.64e25iT - 4.77e51T^{2} \) |
| 19 | \( 1 - 4.61e26T + 5.10e53T^{2} \) |
| 23 | \( 1 + 3.87e28iT - 1.55e57T^{2} \) |
| 29 | \( 1 + 4.99e30iT - 2.63e61T^{2} \) |
| 31 | \( 1 + 6.62e30T + 4.33e62T^{2} \) |
| 37 | \( 1 - 1.25e33T + 7.31e65T^{2} \) |
| 41 | \( 1 + 6.68e33iT - 5.45e67T^{2} \) |
| 43 | \( 1 - 1.15e34T + 4.03e68T^{2} \) |
| 47 | \( 1 - 6.11e34iT - 1.69e70T^{2} \) |
| 53 | \( 1 - 1.29e36iT - 2.62e72T^{2} \) |
| 59 | \( 1 - 6.95e36iT - 2.37e74T^{2} \) |
| 61 | \( 1 + 4.26e37T + 9.63e74T^{2} \) |
| 67 | \( 1 + 1.36e38T + 4.95e76T^{2} \) |
| 71 | \( 1 + 7.56e38iT - 5.66e77T^{2} \) |
| 73 | \( 1 - 7.90e38T + 1.81e78T^{2} \) |
| 79 | \( 1 + 1.56e38T + 5.01e79T^{2} \) |
| 83 | \( 1 + 2.41e40iT - 3.99e80T^{2} \) |
| 89 | \( 1 - 1.04e41iT - 7.48e81T^{2} \) |
| 97 | \( 1 - 3.83e41T + 2.78e83T^{2} \) |
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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
−14.58711914261934392867622660129, −13.78459629945893307285622573386, −11.92326345525225654935099405361, −10.73334854004425740666963008939, −9.309900124241561168184742458765, −7.66823179979168807897408164306, −4.54403831364040380289587996412, −2.96511028068308101683535679188, −2.29049787614503418150105802966, −0.78613974847351663536532908469,
1.51217317833512927964915460531, 4.36710100779523950640896075575, 5.17207589671405116658522715874, 7.45955693210022903325902072110, 8.288740091451856701143249874191, 9.479744847294052402926254237254, 12.85943484082784398595176016185, 14.41301786348410092911618841287, 15.18449048448021277631535427996, 16.70881113589277059568314612530