L(s) = 1 | + 3·2-s + 4·4-s − 6·5-s + 3·8-s − 18·10-s + 3·13-s + 3·16-s + 3·17-s + 9·19-s − 24·20-s + 17·25-s + 9·26-s + 9·29-s + 6·31-s + 6·32-s + 9·34-s − 7·37-s + 27·38-s − 18·40-s + 3·41-s − 43-s + 51·50-s + 12·52-s + 15·53-s + 27·58-s − 24·61-s + 18·62-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 2·4-s − 2.68·5-s + 1.06·8-s − 5.69·10-s + 0.832·13-s + 3/4·16-s + 0.727·17-s + 2.06·19-s − 5.36·20-s + 17/5·25-s + 1.76·26-s + 1.67·29-s + 1.07·31-s + 1.06·32-s + 1.54·34-s − 1.15·37-s + 4.37·38-s − 2.84·40-s + 0.468·41-s − 0.152·43-s + 7.21·50-s + 1.66·52-s + 2.06·53-s + 3.54·58-s − 3.07·61-s + 2.28·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.425555612\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.425555612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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
−9.932337160153731746957062503742, −9.530235682617509298892648856474, −8.595002802637933244445799229005, −8.585368582330651364934799202274, −8.129543667534550246380302483436, −7.58601057620460057814861232738, −7.33736990755634790394001229971, −7.13267098647986541258074519344, −6.25627095719402938198468542649, −6.09240038635763858456755089859, −5.25921481403445240316861549242, −5.20035891424611900075407737986, −4.46935596736090048007741153300, −4.35013173237213191048799836224, −3.81726085036152213733946959978, −3.47542406304217755388431594863, −3.10836248469462041144498871373, −2.80766135364615153005046163132, −1.34542189728648312782016143691, −0.70130375055115347042275108645,
0.70130375055115347042275108645, 1.34542189728648312782016143691, 2.80766135364615153005046163132, 3.10836248469462041144498871373, 3.47542406304217755388431594863, 3.81726085036152213733946959978, 4.35013173237213191048799836224, 4.46935596736090048007741153300, 5.20035891424611900075407737986, 5.25921481403445240316861549242, 6.09240038635763858456755089859, 6.25627095719402938198468542649, 7.13267098647986541258074519344, 7.33736990755634790394001229971, 7.58601057620460057814861232738, 8.129543667534550246380302483436, 8.585368582330651364934799202274, 8.595002802637933244445799229005, 9.530235682617509298892648856474, 9.932337160153731746957062503742