| L(s) = 1 | − 5-s − 3·7-s − 5·11-s + 5·13-s + 4·17-s + 8·19-s + 23-s + 5·25-s − 9·29-s − 31-s + 3·35-s − 12·37-s + 3·41-s + 43-s + 3·47-s + 7·49-s − 4·53-s + 5·55-s − 11·59-s − 7·61-s − 5·65-s − 67-s + 8·71-s − 4·73-s + 15·77-s + 79-s − 83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 1.50·11-s + 1.38·13-s + 0.970·17-s + 1.83·19-s + 0.208·23-s + 25-s − 1.67·29-s − 0.179·31-s + 0.507·35-s − 1.97·37-s + 0.468·41-s + 0.152·43-s + 0.437·47-s + 49-s − 0.549·53-s + 0.674·55-s − 1.43·59-s − 0.896·61-s − 0.620·65-s − 0.122·67-s + 0.949·71-s − 0.468·73-s + 1.70·77-s + 0.112·79-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.186891601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186891601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.51177815108204120694486593573, −10.73820080337191516685457593816, −10.36897861647520783416578977668, −10.36696948266601687657928147189, −9.394197481561256005738976917049, −9.178841475932890800747327845910, −8.841304602998409648703408063507, −7.987049205959040255893155906594, −7.64853519643636250393263359976, −7.42987543268824201292823725330, −6.73076576302961635812917879929, −6.17230312203542733600921981594, −5.56049991448361510819767810295, −5.33980905806650582250778550093, −4.65323200998414081523744658923, −3.66384445606605246918677394080, −3.27892027375685756721337790678, −3.07654832083245814081515633381, −1.86510636850193483578702143647, −0.70996835028244278359339028172,
0.70996835028244278359339028172, 1.86510636850193483578702143647, 3.07654832083245814081515633381, 3.27892027375685756721337790678, 3.66384445606605246918677394080, 4.65323200998414081523744658923, 5.33980905806650582250778550093, 5.56049991448361510819767810295, 6.17230312203542733600921981594, 6.73076576302961635812917879929, 7.42987543268824201292823725330, 7.64853519643636250393263359976, 7.987049205959040255893155906594, 8.841304602998409648703408063507, 9.178841475932890800747327845910, 9.394197481561256005738976917049, 10.36696948266601687657928147189, 10.36897861647520783416578977668, 10.73820080337191516685457593816, 11.51177815108204120694486593573
