L(s) = 1 | + (−1.63 − 1.52i)5-s + 5.27·7-s + 3·9-s − 6.50i·11-s − 3.88i·17-s + 4.35i·19-s − 4·23-s + (0.362 + 4.98i)25-s + (−8.63 − 8.03i)35-s − 11.8·43-s + (−4.91 − 4.56i)45-s + 2.72·47-s + 20.8·49-s + (−9.91 + 10.6i)55-s + 11.2·61-s + ⋯ |
L(s) = 1 | + (−0.732 − 0.680i)5-s + 1.99·7-s + 9-s − 1.96i·11-s − 0.941i·17-s + 0.999i·19-s − 0.834·23-s + (0.0725 + 0.997i)25-s + (−1.45 − 1.35i)35-s − 1.80·43-s + (−0.732 − 0.680i)45-s + 0.397·47-s + 2.97·49-s + (−1.33 + 1.43i)55-s + 1.44·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.926419237\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.926419237\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.63 + 1.52i)T \) |
| 19 | \( 1 - 4.35iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 5.27T + 7T^{2} \) |
| 11 | \( 1 + 6.50iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.88iT - 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 2.72T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.0iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 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
−9.052775334906463461582109981511, −8.229353475042505588992269480327, −8.044363560315186393722789867521, −7.13185191320970534664121945324, −5.79556211607949632014508615145, −5.06489940122860980211844062065, −4.30757168771991243504892240395, −3.46245100425127740208345670894, −1.79235697479113547948783657012, −0.843933468200481199120076252527,
1.55672970096238837123913419689, 2.28449554131356299831204552254, 4.04193100459705623697379320468, 4.44200796323259497350618588944, 5.22699789060008029786820967873, 6.75583999288048772738844609160, 7.26804981491160829385519288835, 7.924197326101635720799229039596, 8.583017920322716614909415929377, 9.886337823919349481154021037626