| L(s) = 1 | + 4.76·3-s − 15.5·7-s − 4.30·9-s − 74.3·21-s + 207.·23-s − 149.·27-s + 306·29-s + 460.·41-s − 30.9·43-s + 643.·47-s − 99.7·49-s + 40.2·61-s + 67.1·63-s + 1.09e3·67-s + 990.·69-s − 594.·81-s − 1.14e3·83-s + 1.45e3·87-s − 1.38e3·89-s − 378·101-s + 1.98e3·103-s + 1.77e3·107-s − 1.97e3·109-s + ⋯ |
| L(s) = 1 | + 0.916·3-s − 0.842·7-s − 0.159·9-s − 0.772·21-s + 1.88·23-s − 1.06·27-s + 1.95·29-s + 1.75·41-s − 0.109·43-s + 1.99·47-s − 0.290·49-s + 0.0844·61-s + 0.134·63-s + 1.99·67-s + 1.72·69-s − 0.815·81-s − 1.51·83-s + 1.79·87-s − 1.65·89-s − 0.372·101-s + 1.89·103-s + 1.59·107-s − 1.73·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.489981020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489981020\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.76T + 27T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 306T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 - 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 643.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.09e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.665100365893924827949272038506, −9.010469382426616817428277542851, −8.344176940099416007143729431144, −7.33014321937830420955061285894, −6.51447420771866517429125727037, −5.46672794870478567095195885314, −4.25024550019534403749302682128, −3.12543284921504462390505219026, −2.54810363095590080831875985583, −0.844886913986003852356878683321,
0.844886913986003852356878683321, 2.54810363095590080831875985583, 3.12543284921504462390505219026, 4.25024550019534403749302682128, 5.46672794870478567095195885314, 6.51447420771866517429125727037, 7.33014321937830420955061285894, 8.344176940099416007143729431144, 9.010469382426616817428277542851, 9.665100365893924827949272038506
