L(s) = 1 | − 3-s + 0.572i·5-s + i·7-s + 9-s − 6.41i·11-s + (0.898 − 3.49i)13-s − 0.572i·15-s − 5.02·17-s − 5.73i·19-s − i·21-s − 2.81·23-s + 4.67·25-s − 27-s − 3.02·29-s + 6.41i·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.256i·5-s + 0.377i·7-s + 0.333·9-s − 1.93i·11-s + (0.249 − 0.968i)13-s − 0.147i·15-s − 1.21·17-s − 1.31i·19-s − 0.218i·21-s − 0.587·23-s + 0.934·25-s − 0.192·27-s − 0.560·29-s + 1.11i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2023367379\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2023367379\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-0.898 + 3.49i)T \) |
good | 5 | \( 1 - 0.572iT - 5T^{2} \) |
| 11 | \( 1 + 6.41iT - 11T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 5.73iT - 19T^{2} \) |
| 23 | \( 1 + 2.81T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2.16iT - 37T^{2} \) |
| 41 | \( 1 - 12.3iT - 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 9.43iT - 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 - 12.9iT - 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 - 2.41iT - 67T^{2} \) |
| 71 | \( 1 - 4.95iT - 71T^{2} \) |
| 73 | \( 1 - 8.71iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 5.80iT - 83T^{2} \) |
| 89 | \( 1 - 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 4.94iT - 97T^{2} \) |
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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
−8.200661793504163585461843695074, −7.04355368231306948479121877725, −6.45171953847495023884323570920, −5.78151338396789169548073898859, −5.17539446168462687074774414710, −4.24239173016219202912867326126, −3.18627139638185565265401798423, −2.61569900666153133013741114797, −1.10399952736346627260511374800, −0.06577767574871217095177396083,
1.60753370412119426124993991967, 2.10693143966625630374385699980, 3.69773002965927922361397641060, 4.41494782597766087150231029483, 4.84532948171780413434544554776, 5.89278010312121497691540595575, 6.67554424005818779983989924512, 7.17071887721678853781068141168, 7.88757451110649210439082336541, 8.875015753893973853965543909517