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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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
−10.01365128586718589913162096684, −9.046701065131420596037523675746, −8.465634149454526514018613232318, −7.37689996600776941010133973889, −6.54783502541087202812385298996, −5.72980615123708746539580485902, −5.07457748482028840930512078021, −3.68453294955042937707686757542, −2.69845957610737187130717186504, −1.69782063543272412998390225850,
0.48391558386687000119496086281, 2.23040345481990869138687480072, 2.90589578051770929202923868807, 4.47998107465535218784934816901, 5.27385005860111024739989216656, 5.91144953094341588494559690297, 6.97716396557392237961911171739, 7.78708472491273961724853956710, 8.830178139484541393255483086836, 9.406342600016678465411350823286