| L(s) = 1 | − 1.77i·3-s + 3.16i·5-s + (0.435 + 2.60i)7-s − 0.160·9-s + (−1.86 − 2.74i)11-s + 13-s + 5.62·15-s + 2.82·17-s + 2.08·19-s + (4.63 − 0.773i)21-s + 7.03·23-s − 5.00·25-s − 5.04i·27-s + 8.40i·29-s + 7.59i·31-s + ⋯ |
| L(s) = 1 | − 1.02i·3-s + 1.41i·5-s + (0.164 + 0.986i)7-s − 0.0534·9-s + (−0.563 − 0.826i)11-s + 0.277·13-s + 1.45·15-s + 0.685·17-s + 0.478·19-s + (1.01 − 0.168i)21-s + 1.46·23-s − 1.00·25-s − 0.971i·27-s + 1.56i·29-s + 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992839041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992839041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.435 - 2.60i)T \) |
| 11 | \( 1 + (1.86 + 2.74i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.77iT - 3T^{2} \) |
| 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 - 8.40iT - 29T^{2} \) |
| 31 | \( 1 - 7.59iT - 31T^{2} \) |
| 37 | \( 1 + 6.03T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 8.16iT - 43T^{2} \) |
| 47 | \( 1 + 8.73iT - 47T^{2} \) |
| 53 | \( 1 - 0.279T + 53T^{2} \) |
| 59 | \( 1 - 4.51iT - 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 - 5.00iT - 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 12.9iT - 97T^{2} \) |
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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
−8.547379891970497093243291406144, −7.59550794956465344775195498222, −6.98474363223603241961989519137, −6.62351699161112677552965478914, −5.59001289379683041672549524404, −5.17583711146265769871104219698, −3.46582522632408126253045693585, −3.02993800029547332312082520370, −2.16924891781583428312870183825, −1.10926218384948815391375592954,
0.67058682523798473816423550590, 1.61005806172993187576108551279, 3.07503277355492663136419126165, 4.10403054023455298317488756974, 4.53339015534816343810686412563, 5.08116795207756614011872175648, 5.86079784875244853139170455917, 7.10855223452714266309270710952, 7.71324057246020258560918149248, 8.386504122511404741253860967583
