L(s) = 1 | + (0.881 − 1.49i)3-s − 1.27i·5-s − 0.100i·7-s + (−1.44 − 2.62i)9-s + 3.88·11-s + 5.91·13-s + (−1.90 − 1.12i)15-s + 2.12i·17-s − i·19-s + (−0.150 − 0.0887i)21-s − 8.22·23-s + 3.36·25-s + (−5.19 − 0.165i)27-s − 7.33i·29-s − 2.91i·31-s + ⋯ |
L(s) = 1 | + (0.509 − 0.860i)3-s − 0.572i·5-s − 0.0380i·7-s + (−0.481 − 0.876i)9-s + 1.17·11-s + 1.64·13-s + (−0.492 − 0.291i)15-s + 0.514i·17-s − 0.229i·19-s + (−0.0327 − 0.0193i)21-s − 1.71·23-s + 0.672·25-s + (−0.999 − 0.0318i)27-s − 1.36i·29-s − 0.523i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39868 - 1.41360i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39868 - 1.41360i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.881 + 1.49i)T \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + 1.27iT - 5T^{2} \) |
| 7 | \( 1 + 0.100iT - 7T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 2.12iT - 17T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 + 7.33iT - 29T^{2} \) |
| 31 | \( 1 + 2.91iT - 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 + 0.0327iT - 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 - 13.7iT - 53T^{2} \) |
| 59 | \( 1 + 8.10T + 59T^{2} \) |
| 61 | \( 1 + 4.04T + 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 3.62T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 - 0.457iT - 79T^{2} \) |
| 83 | \( 1 + 2.00T + 83T^{2} \) |
| 89 | \( 1 + 11.0iT - 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.644175510771235775419310990321, −8.784851958077018712131803867023, −8.384625909697304995069549815966, −7.42759560545548472017707318760, −6.28532722322545646887180632610, −5.95344096617961520971220183205, −4.24410958149763909621065326271, −3.55005362014844263466170330780, −2.00535390197232015205930648822, −0.993692517491524582644342604973,
1.70567807447553766949661450604, 3.25258643703056420996594470425, 3.76108797369090193809139346608, 4.87307741163472256220420723287, 6.05894612540078462043458596979, 6.77934598829867942407902811830, 8.024738563335303134384479921870, 8.727677001755367969755824563737, 9.427941614444536179620742688344, 10.31027420527838716115419545763