L(s) = 1 | + 1.27·5-s + 2.45i·7-s + 4.60i·11-s + 5.38i·13-s + 2.30·17-s + (0.0756 − 4.35i)19-s − 3.46i·23-s − 3.38·25-s + 4.46i·29-s + 0.666·31-s + 3.11i·35-s + 4.23i·37-s + 7.19i·41-s − 1.91i·43-s − 1.97i·47-s + ⋯ |
L(s) = 1 | + 0.568·5-s + 0.927i·7-s + 1.38i·11-s + 1.49i·13-s + 0.559·17-s + (0.0173 − 0.999i)19-s − 0.723i·23-s − 0.677·25-s + 0.828i·29-s + 0.119·31-s + 0.526i·35-s + 0.696i·37-s + 1.12i·41-s − 0.291i·43-s − 0.288i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.703988097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.703988097\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.0756 + 4.35i)T \) |
good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 2.45iT - 7T^{2} \) |
| 11 | \( 1 - 4.60iT - 11T^{2} \) |
| 13 | \( 1 - 5.38iT - 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 4.46iT - 29T^{2} \) |
| 31 | \( 1 - 0.666T + 31T^{2} \) |
| 37 | \( 1 - 4.23iT - 37T^{2} \) |
| 41 | \( 1 - 7.19iT - 41T^{2} \) |
| 43 | \( 1 + 1.91iT - 43T^{2} \) |
| 47 | \( 1 + 1.97iT - 47T^{2} \) |
| 53 | \( 1 - 7.96iT - 53T^{2} \) |
| 59 | \( 1 + 0.672T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 7.51iT - 83T^{2} \) |
| 89 | \( 1 + 1.80iT - 89T^{2} \) |
| 97 | \( 1 + 6.69iT - 97T^{2} \) |
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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
−8.668041962416408909915840204560, −7.60424826418677366766398308580, −6.91571839699363663180566542404, −6.36632040018697382403371400539, −5.53150835199504416083302546427, −4.75675873667330227051118399472, −4.22181051267444130995513981086, −2.91724461413319220039126286586, −2.18769857611598658113655447640, −1.48848877050113105216222621849,
0.44231826117443286269841321657, 1.31828020741417426840712902291, 2.55051885715568137600474603156, 3.53536500693029327482188626201, 3.90102088612571112886931082710, 5.29049992546898790249551343907, 5.70205518601415769090707814538, 6.27327988392514178075342331646, 7.36504086973600838320096839995, 7.891838571781690124227040751299