L(s) = 1 | + (−0.707 − 1.22i)3-s + (0.5 − 0.866i)4-s − i·5-s + (−0.499 + 0.866i)9-s − 1.41·12-s + (−0.965 + 0.258i)13-s + (−1.22 + 0.707i)15-s + (−0.499 − 0.866i)16-s + (−0.258 + 0.448i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (0.258 + 0.448i)23-s − 25-s + i·31-s + (0.5 + 0.866i)36-s + (1.67 − 0.965i)37-s + ⋯ |
L(s) = 1 | + (−0.707 − 1.22i)3-s + (0.5 − 0.866i)4-s − i·5-s + (−0.499 + 0.866i)9-s − 1.41·12-s + (−0.965 + 0.258i)13-s + (−1.22 + 0.707i)15-s + (−0.499 − 0.866i)16-s + (−0.258 + 0.448i)17-s + (−0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (0.258 + 0.448i)23-s − 25-s + i·31-s + (0.5 + 0.866i)36-s + (1.67 − 0.965i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7699952923\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7699952923\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 + (0.965 - 0.258i)T \) |
| 31 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.93T + T^{2} \) |
| 59 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.517iT - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.094648526852402433530552590629, −7.83601458231890239735638887189, −7.30317102368891381796007105369, −6.44714794099345335471252157667, −5.86853017789903756906387418641, −5.11896233763906585033289100549, −4.27965823257536400864728169352, −2.41197655677768046700255391762, −1.66822382485897813806817726776, −0.57446071008620327258759194942,
2.41187827335395227701244712834, 3.06238446534896681538535282432, 4.19914062305828782896550706239, 4.63224685801687549051381984427, 6.00085294561658541941478964753, 6.41213788295742204354607209833, 7.54670940188196031382830373201, 7.955807066878635686117839679986, 9.278398151920701907078100757614, 9.846873738844057689596061863259