L(s) = 1 | − 5-s + 5·11-s − 4·17-s + 8·19-s + 4·23-s − 4·25-s − 5·29-s − 3·31-s + 4·37-s − 2·43-s − 6·47-s − 9·53-s − 5·55-s + 11·59-s − 6·61-s + 2·67-s − 2·71-s − 10·73-s + 3·79-s + 7·83-s + 4·85-s − 6·89-s − 8·95-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.50·11-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 0.928·29-s − 0.538·31-s + 0.657·37-s − 0.304·43-s − 0.875·47-s − 1.23·53-s − 0.674·55-s + 1.43·59-s − 0.768·61-s + 0.244·67-s − 0.237·71-s − 1.17·73-s + 0.337·79-s + 0.768·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−15.48539879222369, −14.80467342040636, −14.61151564336799, −13.81849874391728, −13.43799253740963, −12.84674388914028, −12.13515122043158, −11.66368263739680, −11.29108760521468, −10.88478650629165, −9.831762379154666, −9.489652145368360, −9.068117227308543, −8.390681300542595, −7.695071747422007, −7.183945313807063, −6.673666685517461, −6.015038279952636, −5.326798928299463, −4.665594484578498, −3.936634648982961, −3.515268373055119, −2.749358322382546, −1.718116151707914, −1.124697814261402, 0,
1.124697814261402, 1.718116151707914, 2.749358322382546, 3.515268373055119, 3.936634648982961, 4.665594484578498, 5.326798928299463, 6.015038279952636, 6.673666685517461, 7.183945313807063, 7.695071747422007, 8.390681300542595, 9.068117227308543, 9.489652145368360, 9.831762379154666, 10.88478650629165, 11.29108760521468, 11.66368263739680, 12.13515122043158, 12.84674388914028, 13.43799253740963, 13.81849874391728, 14.61151564336799, 14.80467342040636, 15.48539879222369