L(s) = 1 | − 2·4-s + 3.60·5-s − 3·9-s + 4·16-s + 3.60·19-s − 7.21·20-s − 23-s + 7.99·25-s − 5·29-s − 10.8·31-s + 6·36-s + 7.21·41-s − 9·43-s − 10.8·45-s − 3.60·47-s + 11·53-s − 14.4·59-s − 8·64-s + 10.8·73-s − 7.21·76-s + 15·79-s + 14.4·80-s + 9·81-s − 18.0·83-s − 3.60·89-s + 2·92-s + 12.9·95-s + ⋯ |
L(s) = 1 | − 4-s + 1.61·5-s − 9-s + 16-s + 0.827·19-s − 1.61·20-s − 0.208·23-s + 1.59·25-s − 0.928·29-s − 1.94·31-s + 36-s + 1.12·41-s − 1.37·43-s − 1.61·45-s − 0.525·47-s + 1.51·53-s − 1.87·59-s − 64-s + 1.26·73-s − 0.827·76-s + 1.68·79-s + 1.61·80-s + 81-s − 1.97·83-s − 0.382·89-s + 0.208·92-s + 1.33·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53852581771901764728460040806, −6.62985241132862204882733860439, −5.67900318941581111724210013030, −5.59943755236485451905141829373, −4.91084457216676906447258676687, −3.82872584793057985569687657438, −3.09308938464490423714769388237, −2.16578795526503441363499567869, −1.30412475029278041531960892734, 0,
1.30412475029278041531960892734, 2.16578795526503441363499567869, 3.09308938464490423714769388237, 3.82872584793057985569687657438, 4.91084457216676906447258676687, 5.59943755236485451905141829373, 5.67900318941581111724210013030, 6.62985241132862204882733860439, 7.53852581771901764728460040806