| L(s) = 1 | − 1.59i·2-s + 13.4·4-s − 13.8i·5-s + 2.62·7-s − 46.9i·8-s − 22.1·10-s + 139. i·11-s − 147.·13-s − 4.19i·14-s + 140.·16-s + 173. i·17-s + 512.·19-s − 186. i·20-s + 222.·22-s + 3.56i·23-s + ⋯ |
| L(s) = 1 | − 0.398i·2-s + 0.840·4-s − 0.555i·5-s + 0.0536·7-s − 0.734i·8-s − 0.221·10-s + 1.15i·11-s − 0.874·13-s − 0.0214i·14-s + 0.548·16-s + 0.598i·17-s + 1.42·19-s − 0.467i·20-s + 0.460·22-s + 0.00674i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.792077338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.792077338\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 + 1.59iT - 16T^{2} \) |
| 5 | \( 1 + 13.8iT - 625T^{2} \) |
| 7 | \( 1 - 2.62T + 2.40e3T^{2} \) |
| 11 | \( 1 - 139. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 147.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 173. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 512.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 3.56iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 323. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.12e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.66e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 176. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 715.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.06e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.69e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 4.03e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.98e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 440. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.68e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.99e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.46e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.04e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.89e3T + 8.85e7T^{2} \) |
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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
−9.973599416043794056744463431700, −9.679782011188408232633370750125, −8.298436847542441431892832954278, −7.37673640952372720836721220795, −6.66294383637626976096998928324, −5.38091424259028627721996832447, −4.45116487979388412999068342514, −3.10661667831740189294043312037, −2.02958363079068856992956618665, −0.949698730429559784918664567555,
0.917128455546556861096898566991, 2.56255070834979476044638423762, 3.21170822603824967228574532140, 4.90520653338950931339745040655, 5.87445857659398071115732088358, 6.72548762914693077774373649236, 7.52747538459118495329430485375, 8.277075792565979297334559444630, 9.530536633551666393493749659223, 10.36423380226271179210939839799
