L(s) = 1 | − 3.23·3-s − 0.763·7-s + 7.47·9-s + 2.47·21-s + 5.70·23-s − 14.4·27-s − 6·29-s − 4.47·41-s − 11.2·43-s − 13.7·47-s − 6.41·49-s − 13.4·61-s − 5.70·63-s + 8.18·67-s − 18.4·69-s + 24.4·81-s + 17.7·83-s + 19.4·87-s + 6·89-s − 18·101-s − 20.1·103-s + 6.29·107-s + 13.4·109-s + ⋯ |
L(s) = 1 | − 1.86·3-s − 0.288·7-s + 2.49·9-s + 0.539·21-s + 1.19·23-s − 2.78·27-s − 1.11·29-s − 0.698·41-s − 1.71·43-s − 1.99·47-s − 0.916·49-s − 1.71·61-s − 0.719·63-s + 0.999·67-s − 2.22·69-s + 2.71·81-s + 1.94·83-s + 2.08·87-s + 0.635·89-s − 1.79·101-s − 1.98·103-s + 0.608·107-s + 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 7 | \( 1 + 0.763T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 13.7T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.03737833035502482587801863851, −9.300582179241337954509875615970, −7.924778022804316373882635712297, −6.85684343518019557369700046443, −6.38706411435082990241653706437, −5.32430888987114368555108609273, −4.77612870875324925388058545659, −3.47910078778398930143432965745, −1.52217958752794764656562061729, 0,
1.52217958752794764656562061729, 3.47910078778398930143432965745, 4.77612870875324925388058545659, 5.32430888987114368555108609273, 6.38706411435082990241653706437, 6.85684343518019557369700046443, 7.924778022804316373882635712297, 9.300582179241337954509875615970, 10.03737833035502482587801863851