L(s) = 1 | + (−2 − i)5-s − 3·9-s + 4·13-s + 8i·17-s + (3 + 4i)25-s − 10i·29-s + 12·37-s + 10·41-s + (6 + 3i)45-s + 7·49-s + 4·53-s + 10i·61-s + (−8 − 4i)65-s + 16i·73-s + 9·81-s + ⋯ |
L(s) = 1 | + (−0.894 − 0.447i)5-s − 9-s + 1.10·13-s + 1.94i·17-s + (0.600 + 0.800i)25-s − 1.85i·29-s + 1.97·37-s + 1.56·41-s + (0.894 + 0.447i)45-s + 49-s + 0.549·53-s + 1.28i·61-s + (−0.992 − 0.496i)65-s + 1.87i·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239894465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239894465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 8iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 12T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.615574314923288353865570939568, −8.577254095764506108823686427941, −8.308118398786823912895194295791, −7.48212053139456271821073999341, −6.09564992246313989759138835603, −5.79976791203005980635678570278, −4.27959776292931403413698728579, −3.83337270907240995283737323352, −2.54437320042756666156764161848, −0.953118674934607983376915718507,
0.71385340019705987594688371827, 2.66161351336519612778498726509, 3.34422504374411164713378759684, 4.43026559039929634531679062070, 5.43442945106864257347401247849, 6.37581671780996700173250747470, 7.25242257584491243425823245549, 7.961190862648714773199738838735, 8.835687006138474121852220006503, 9.429711402717261654722650621073