| L(s) = 1 | − 8·5-s − 3·9-s − 36·17-s − 2·25-s + 8·29-s + 144·37-s − 36·41-s + 24·45-s + 50·49-s − 88·53-s − 144·61-s + 164·73-s + 9·81-s + 288·85-s − 252·89-s + 220·97-s − 184·101-s − 288·109-s − 252·113-s + 194·121-s + 344·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 1/3·9-s − 2.11·17-s − 0.0799·25-s + 8/29·29-s + 3.89·37-s − 0.878·41-s + 8/15·45-s + 1.02·49-s − 1.66·53-s − 2.36·61-s + 2.24·73-s + 1/9·81-s + 3.38·85-s − 2.83·89-s + 2.26·97-s − 1.82·101-s − 2.64·109-s − 2.23·113-s + 1.60·121-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.441·145-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2738871721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2738871721\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 670 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2690 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8546 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8354 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8594 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 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
−10.51441758882441606494961169666, −9.683812991124919364960528717269, −9.497887904497105797759077795865, −9.123393359606238580007183979689, −8.329739780746190286146620614345, −8.303843887130542466431687944384, −7.78453576834799799880328189499, −7.46080457157381997766701586741, −6.91159832148606095664768353703, −6.23874225324380320766477949299, −6.22909918451600164528141805674, −5.39929534159301542481198665948, −4.65001419196546255673891595530, −4.37006153693629314802574784735, −4.05663567865512245022222782329, −3.42116533377291872565787190598, −2.69530604468412537482281238030, −2.28913126131634597050883117703, −1.22954929847052952831888931752, −0.19259348445552987089425141209,
0.19259348445552987089425141209, 1.22954929847052952831888931752, 2.28913126131634597050883117703, 2.69530604468412537482281238030, 3.42116533377291872565787190598, 4.05663567865512245022222782329, 4.37006153693629314802574784735, 4.65001419196546255673891595530, 5.39929534159301542481198665948, 6.22909918451600164528141805674, 6.23874225324380320766477949299, 6.91159832148606095664768353703, 7.46080457157381997766701586741, 7.78453576834799799880328189499, 8.303843887130542466431687944384, 8.329739780746190286146620614345, 9.123393359606238580007183979689, 9.497887904497105797759077795865, 9.683812991124919364960528717269, 10.51441758882441606494961169666
