L(s) = 1 | + (−1.24 + 1.20i)3-s − 2·4-s − 3.58i·5-s + 5.15·7-s + (0.0927 − 2.99i)9-s + (2.48 − 2.41i)12-s + (4.32 + 4.45i)15-s + 4·16-s − 7.68i·17-s − 2.30·19-s + 7.17i·20-s + (−6.41 + 6.22i)21-s − 7.85·25-s + (3.5 + 3.84i)27-s − 10.3·28-s + 3.64i·29-s + ⋯ |
L(s) = 1 | + (−0.717 + 0.696i)3-s − 4-s − 1.60i·5-s + 1.95·7-s + (0.0309 − 0.999i)9-s + (0.717 − 0.696i)12-s + (1.11 + 1.15i)15-s + 16-s − 1.86i·17-s − 0.528·19-s + 1.60i·20-s + (−1.40 + 1.35i)21-s − 1.57·25-s + (0.673 + 0.739i)27-s − 1.95·28-s + 0.677i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.789212 - 0.334070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.789212 - 0.334070i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.24 - 1.20i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.58iT - 5T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.68iT - 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3.64iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 0.0624iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.4iT - 53T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 7.68iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.33815347320516769639360902408, −11.72680220167313986597668436563, −10.62958286648556421964805883303, −9.274662667136823741138594174031, −8.804254986455005981271907048541, −7.73243397171610173664706880439, −5.45614113122990597672285857486, −4.87970349241540486350579246227, −4.32469803135168192830013387671, −1.01085268182840106376343442750,
1.90759366497427453963034178453, 4.08726542167024137495585613196, 5.36430593724127608616428479917, 6.48410664301203497605651106638, 7.79910666977781083640403494494, 8.341750769084404328942501304358, 10.26641059138070064135458963944, 10.88827535187030217298960639791, 11.65298783091615526064075650452, 12.84049321681352322028510340244