L(s) = 1 | + (2.09 − 3.62i)5-s + (1.62 − 2.09i)7-s + (2.59 − 1.5i)11-s + 2.44i·13-s + (−0.507 − 0.878i)17-s + (0.878 + 0.507i)19-s + (3.67 + 2.12i)23-s + (−6.24 − 10.8i)25-s + 1.24i·29-s + (−4.86 + 2.80i)31-s + (−4.18 − 10.2i)35-s + (−4.12 + 7.13i)37-s + 2.02·41-s − 8.24·43-s + (0.507 − 0.878i)47-s + ⋯ |
L(s) = 1 | + (0.935 − 1.61i)5-s + (0.612 − 0.790i)7-s + (0.783 − 0.452i)11-s + 0.679i·13-s + (−0.123 − 0.213i)17-s + (0.201 + 0.116i)19-s + (0.766 + 0.442i)23-s + (−1.24 − 2.16i)25-s + 0.230i·29-s + (−0.873 + 0.504i)31-s + (−0.706 − 1.73i)35-s + (−0.677 + 1.17i)37-s + 0.316·41-s − 1.25·43-s + (0.0739 − 0.128i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.069383237\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.069383237\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.62 + 2.09i)T \) |
good | 5 | \( 1 + (-2.09 + 3.62i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.507 + 0.878i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (4.86 - 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + (-0.507 + 0.878i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.07 + 0.621i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.76iT - 97T^{2} \) |
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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
−9.609119064414538728171807297627, −8.935190081598548881330113736956, −8.406906099513226626161571374765, −7.25111696833699606411365189859, −6.31004615083620550137648153659, −5.23203400396574545207754618534, −4.70309817167901008740532944520, −3.64749594222050520309198451140, −1.79591602732710900154590549549, −1.05064315691622734939393738901,
1.83168900885703679754039013027, 2.64813632942778248163509947592, 3.70567734099486976384918283831, 5.17882138537958431721563387490, 5.92338233099151805691132945182, 6.75898156283073763165416656913, 7.43970725803390592415027321353, 8.587849648663850400908098381263, 9.459271655557506711624553540945, 10.13956023448837574019595045952