L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s − 2·11-s + 5·16-s − 4·18-s + 4·22-s + 16·23-s − 10·25-s + 4·29-s − 6·32-s + 6·36-s + 4·37-s − 8·43-s − 6·44-s − 32·46-s + 20·50-s + 28·53-s − 8·58-s + 7·64-s + 8·67-s − 8·72-s − 8·74-s − 32·79-s − 5·81-s + 16·86-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s − 0.603·11-s + 5/4·16-s − 0.942·18-s + 0.852·22-s + 3.33·23-s − 2·25-s + 0.742·29-s − 1.06·32-s + 36-s + 0.657·37-s − 1.21·43-s − 0.904·44-s − 4.71·46-s + 2.82·50-s + 3.84·53-s − 1.05·58-s + 7/8·64-s + 0.977·67-s − 0.942·72-s − 0.929·74-s − 3.60·79-s − 5/9·81-s + 1.72·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.093366089\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093366089\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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
−10.15102943081488083975851052662, −9.820477154514011232455265934263, −9.085071669556111781842938306281, −8.997466491481587343659289507775, −8.326665340597870716505581679439, −8.323512168724819106043694919347, −7.51717193146548534533156801639, −7.12777322153850334744972367874, −7.11495213396485370016657674131, −6.57607578848459060132495536320, −5.74322054719044137467138824659, −5.66182867008759703605509091147, −4.93923468248198102518591675272, −4.44963573958408879388959489379, −3.78191894378239826821174074022, −3.12631326230837995645522018306, −2.65856383754343291430511836413, −2.05342451150648383206258973081, −1.28019551589000328596134817038, −0.65260617905073516627830173437,
0.65260617905073516627830173437, 1.28019551589000328596134817038, 2.05342451150648383206258973081, 2.65856383754343291430511836413, 3.12631326230837995645522018306, 3.78191894378239826821174074022, 4.44963573958408879388959489379, 4.93923468248198102518591675272, 5.66182867008759703605509091147, 5.74322054719044137467138824659, 6.57607578848459060132495536320, 7.11495213396485370016657674131, 7.12777322153850334744972367874, 7.51717193146548534533156801639, 8.323512168724819106043694919347, 8.326665340597870716505581679439, 8.997466491481587343659289507775, 9.085071669556111781842938306281, 9.820477154514011232455265934263, 10.15102943081488083975851052662