| L(s) = 1 | − 3·3-s + (−6.59 + 9.02i)5-s + 28.2i·7-s + 9·9-s − 23.3i·11-s + 7.81·13-s + (19.7 − 27.0i)15-s − 54.9i·17-s − 137. i·19-s − 84.8i·21-s + 58.1i·23-s + (−37.9 − 119. i)25-s − 27·27-s − 108. i·29-s − 117.·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.590 + 0.807i)5-s + 1.52i·7-s + 0.333·9-s − 0.641i·11-s + 0.166·13-s + (0.340 − 0.466i)15-s − 0.783i·17-s − 1.65i·19-s − 0.881i·21-s + 0.527i·23-s + (−0.303 − 0.952i)25-s − 0.192·27-s − 0.691i·29-s − 0.680·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.204705461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204705461\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + (6.59 - 9.02i)T \) |
| good | 7 | \( 1 - 28.2iT - 343T^{2} \) |
| 11 | \( 1 + 23.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 7.81T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 137. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 58.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 108. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 62.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 326. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 61.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 641. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 818. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 669.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 14.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 69.1iT - 9.12e5T^{2} \) |
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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
−9.509382247496534834160637482846, −8.970660575696004040168738601056, −7.913981343708345733344514051576, −7.05965253424353100914879663812, −6.15207232749671642281571855552, −5.48062338467604103336826041378, −4.42713033866703087666520064871, −3.11003139919101919996935569397, −2.36766306019893542050308799776, −0.51970014426095038170418484659,
0.72537124547935571739210808468, 1.65577485672724815242347972606, 3.76831220741109479209845304427, 4.14510289072269305541871130640, 5.16062612240399524763081909221, 6.18989360684978095382292079312, 7.25082200962248805795338789787, 7.77513157891826871843683373024, 8.707414101963792870360073387441, 9.830927970358815925071629848617
