L(s) = 1 | + (1 − i)2-s − 2i·4-s + (−1 − 2i)5-s + (−2 − 2i)8-s + (−3 − i)10-s + (5 + 5i)13-s − 4·16-s + (3 − 3i)17-s + (−4 + 2i)20-s + (−3 + 4i)25-s + 10·26-s + 10i·29-s + (−4 + 4i)32-s − 6i·34-s + (5 − 5i)37-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + (−0.447 − 0.894i)5-s + (−0.707 − 0.707i)8-s + (−0.948 − 0.316i)10-s + (1.38 + 1.38i)13-s − 16-s + (0.727 − 0.727i)17-s + (−0.894 + 0.447i)20-s + (−0.600 + 0.800i)25-s + 1.96·26-s + 1.85i·29-s + (−0.707 + 0.707i)32-s − 1.02i·34-s + (0.821 − 0.821i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03947 - 1.13742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03947 - 1.13742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-5 - 5i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 10iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−12.28027943639995197037816891852, −11.64321854713640064014423908993, −10.72462583286486581610546272138, −9.376833417237356158778943578610, −8.663247039267933374849409520880, −7.01642705011643884916732879043, −5.67686025332234930635768142191, −4.55360371376143708591131710061, −3.48825090447479168940569400022, −1.43392854484617219200441845323,
3.03351974073920739973609272382, 4.00594085505945662383360458095, 5.68521253712848275746432641839, 6.46953689600195161302849812082, 7.80049789354900015256841351104, 8.337056271821739998998959790353, 10.10198157549735272147512258998, 11.12592721398720404943772405219, 12.04316356748259253012067595730, 13.16047999006397663681282872416