L(s) = 1 | − 3.69·5-s + (−2.41 − 1.08i)7-s + 3.41i·11-s − 5.22i·13-s + 6.75·17-s − 2.16i·19-s + 6.24i·23-s + 8.65·25-s + 2.58i·29-s + 10.4i·31-s + (8.92 + 4i)35-s + 4·37-s − 0.634·41-s + 6.48·43-s + 3.06·47-s + ⋯ |
L(s) = 1 | − 1.65·5-s + (−0.912 − 0.409i)7-s + 1.02i·11-s − 1.44i·13-s + 1.63·17-s − 0.496i·19-s + 1.30i·23-s + 1.73·25-s + 0.480i·29-s + 1.87i·31-s + (1.50 + 0.676i)35-s + 0.657·37-s − 0.0990·41-s + 0.988·43-s + 0.446·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9008448588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9008448588\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.41 + 1.08i)T \) |
good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.22iT - 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 2.16iT - 19T^{2} \) |
| 23 | \( 1 - 6.24iT - 23T^{2} \) |
| 29 | \( 1 - 2.58iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 0.634T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 7.41iT - 71T^{2} \) |
| 73 | \( 1 - 0.896iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 + 9.55iT - 97T^{2} \) |
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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
−10.14062295500920783419013651232, −9.258046648169229382065749173310, −8.122960301035285560816013737872, −7.49028652611995953143605982657, −7.03364880079058188075518393242, −5.65471482010416723415736862031, −4.69171109617430507315947290212, −3.53777298311800705873474280533, −3.13112036818236257647186000002, −0.893349835463759135135995202586,
0.60003355939170464125532095153, 2.68946357983504815647157513133, 3.72083146525629843281771430935, 4.27423460843522577464851501161, 5.73647404836021751665929100425, 6.49712066334368651965745952068, 7.53307480794507235041620197291, 8.158313626258099983697607387883, 8.996010520395629241624250123912, 9.809843371346035329703879429269