| L(s) = 1 | + 3-s − 4·5-s − 3·7-s + 9-s − 2·11-s − 4·15-s + 6·17-s + 4·19-s − 3·21-s − 8·23-s + 11·25-s + 27-s − 4·29-s − 31-s − 2·33-s + 12·35-s − 2·37-s − 12·41-s + 7·43-s − 4·45-s + 6·47-s + 2·49-s + 6·51-s + 4·53-s + 8·55-s + 4·57-s − 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 1.03·15-s + 1.45·17-s + 0.917·19-s − 0.654·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s − 0.742·29-s − 0.179·31-s − 0.348·33-s + 2.02·35-s − 0.328·37-s − 1.87·41-s + 1.06·43-s − 0.596·45-s + 0.875·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s + 1.07·55-s + 0.529·57-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
| 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
−16.02502115026492, −15.58934743083440, −15.46230730299284, −14.50318454391088, −14.16203960056016, −13.42953772592130, −12.78023815096967, −12.24349556168667, −11.96736591221416, −11.28763729956495, −10.49370227405814, −9.961488693874159, −9.501364194881024, −8.619687772218504, −8.155549604750143, −7.482546849963594, −7.358622201231778, −6.436773250060058, −5.610956188308989, −4.933694396108702, −3.917295428350849, −3.573240146546984, −3.167240281625602, −2.210119082312109, −0.8813329833748777, 0,
0.8813329833748777, 2.210119082312109, 3.167240281625602, 3.573240146546984, 3.917295428350849, 4.933694396108702, 5.610956188308989, 6.436773250060058, 7.358622201231778, 7.482546849963594, 8.155549604750143, 8.619687772218504, 9.501364194881024, 9.961488693874159, 10.49370227405814, 11.28763729956495, 11.96736591221416, 12.24349556168667, 12.78023815096967, 13.42953772592130, 14.16203960056016, 14.50318454391088, 15.46230730299284, 15.58934743083440, 16.02502115026492
