L(s) = 1 | − 2.58·3-s − 1.48i·5-s − i·7-s + 3.66·9-s − i·11-s + (1.95 + 3.03i)13-s + 3.82i·15-s − 1.41·17-s − 6.93i·19-s + 2.58i·21-s + 5.98·23-s + 2.80·25-s − 1.71·27-s − 5.74·29-s − 0.366i·31-s + ⋯ |
L(s) = 1 | − 1.49·3-s − 0.662i·5-s − 0.377i·7-s + 1.22·9-s − 0.301i·11-s + (0.541 + 0.840i)13-s + 0.988i·15-s − 0.343·17-s − 1.59i·19-s + 0.563i·21-s + 1.24·23-s + 0.560·25-s − 0.330·27-s − 1.06·29-s − 0.0658i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028182661\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028182661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.95 - 3.03i)T \) |
good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 1.48iT - 5T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6.93iT - 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 + 0.366iT - 31T^{2} \) |
| 37 | \( 1 - 1.67iT - 37T^{2} \) |
| 41 | \( 1 - 7.22iT - 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 - 8.60iT - 47T^{2} \) |
| 53 | \( 1 - 4.62T + 53T^{2} \) |
| 59 | \( 1 + 5.30iT - 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 4.17iT - 67T^{2} \) |
| 71 | \( 1 - 6.38iT - 71T^{2} \) |
| 73 | \( 1 - 6.54iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 3.21iT - 83T^{2} \) |
| 89 | \( 1 - 2.87iT - 89T^{2} \) |
| 97 | \( 1 + 8.69iT - 97T^{2} \) |
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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
−8.459957538932524448284496206156, −7.29441064586854384512691643827, −6.76681924366364058475850029326, −6.13070493695247838215763412729, −5.25233244121434451845807147987, −4.73874179975655884648510625856, −4.07894967665179892333131127482, −2.79184632251705306524190204232, −1.32884676682689219838978082674, −0.59041271833281284010855180913,
0.76406610001856539309252913343, 1.99579707870906622284651374999, 3.20130952303343700839666333563, 4.03689536857019773621192522636, 5.15495009995737406730773239471, 5.59851998269406016857902083778, 6.24397807881045958126896530338, 6.95831386766999334390095225972, 7.59877631423137061000824710366, 8.560365554667694499969986535261