| L(s) = 1 | − 3-s + 9-s − 6·11-s + 5·13-s − 6·17-s + 5·19-s − 6·23-s − 27-s − 6·29-s − 31-s + 6·33-s − 2·37-s − 5·39-s − 43-s − 6·47-s + 6·51-s + 12·53-s − 5·57-s − 6·59-s + 13·61-s + 11·67-s + 6·69-s + 2·73-s − 8·79-s + 81-s + 6·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 0.179·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s − 0.152·43-s − 0.875·47-s + 0.840·51-s + 1.64·53-s − 0.662·57-s − 0.781·59-s + 1.66·61-s + 1.34·67-s + 0.722·69-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.56408312303212, −13.94213956873909, −13.28472936323399, −13.25937022594090, −12.72837353274949, −11.89063581939062, −11.53490097197637, −10.91516978279511, −10.72000961802173, −10.01415924908824, −9.613227637066306, −8.820025413364233, −8.344538929216669, −7.851416658571146, −7.256593655407632, −6.691434088566023, −6.053573462854549, −5.479320955394333, −5.214549328857454, −4.379849736484535, −3.801510771022759, −3.178753328126813, −2.291042289879081, −1.820468025319226, −0.7612954584705766, 0,
0.7612954584705766, 1.820468025319226, 2.291042289879081, 3.178753328126813, 3.801510771022759, 4.379849736484535, 5.214549328857454, 5.479320955394333, 6.053573462854549, 6.691434088566023, 7.256593655407632, 7.851416658571146, 8.344538929216669, 8.820025413364233, 9.613227637066306, 10.01415924908824, 10.72000961802173, 10.91516978279511, 11.53490097197637, 11.89063581939062, 12.72837353274949, 13.25937022594090, 13.28472936323399, 13.94213956873909, 14.56408312303212
