L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + (0.866 + 0.500i)10-s + (−0.133 + 0.5i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.965 + 0.258i)20-s + 1.00i·25-s + 0.517i·26-s + (−0.965 + 1.67i)29-s + (0.258 − 0.965i)32-s + (−1.49 − 0.866i)34-s + (1.36 − 1.36i)37-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + (0.866 + 0.500i)10-s + (−0.133 + 0.5i)13-s + (0.500 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (0.965 + 0.258i)20-s + 1.00i·25-s + 0.517i·26-s + (−0.965 + 1.67i)29-s + (0.258 − 0.965i)32-s + (−1.49 − 0.866i)34-s + (1.36 − 1.36i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.203874691\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.203874691\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 41 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 0.517T + T^{2} \) |
| 97 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.585781410086267703735639241071, −9.106448107606240040960735728627, −7.61035683272436962632606430126, −6.88291827769189969699967262845, −6.35464189172344624058535781347, −5.35516832768796322301801023310, −4.65723356416275828324993994770, −3.52442965882191482583744149730, −2.62495250444300182537504969738, −1.76460930281198264879282382809,
1.71537300918316557586658568835, 2.63780158461373181775080698332, 3.96194431360034616461448886154, 4.61893406479561407886260633076, 5.60420743905689515757235479182, 6.13721332240979299709863113527, 6.97309417819355674225366092139, 8.110674161046732775698014627719, 8.550814394519954855215459324942, 9.704561092003897085859492748041