| L(s) = 1 | + 12·4-s − 9·9-s − 42·11-s + 80·16-s − 32·19-s − 334·29-s + 20·31-s − 108·36-s − 336·41-s − 504·44-s − 49·49-s − 976·59-s + 56·61-s + 192·64-s − 570·71-s − 384·76-s + 938·79-s + 81·81-s − 648·89-s + 378·99-s + 68·101-s + 2.93e3·109-s − 4.00e3·116-s − 1.33e3·121-s + 240·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s − 1.15·11-s + 5/4·16-s − 0.386·19-s − 2.13·29-s + 0.115·31-s − 1/2·36-s − 1.27·41-s − 1.72·44-s − 1/7·49-s − 2.15·59-s + 0.117·61-s + 3/8·64-s − 0.952·71-s − 0.579·76-s + 1.33·79-s + 1/9·81-s − 0.771·89-s + 0.383·99-s + 0.0669·101-s + 2.57·109-s − 3.20·116-s − 1.00·121-s + 0.173·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.910922805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910922805\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 21 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3818 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9342 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 23709 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 167 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 83617 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 149605 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 47646 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 264630 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 488 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 333563 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 285 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 75790 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 469 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 978738 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1812350 T^{2} + p^{6} T^{4} \) |
| 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
−10.95434016999755428996981208780, −10.21420790210294167924953949616, −10.09822296234234106787534183967, −9.223980873564047010625797149322, −9.034035468909475440393734547129, −8.228796414578987652865381425721, −7.948282652397653849486575767798, −7.33274278909123766119537041455, −7.28808761962150163831461415376, −6.35826795053767554196357805109, −6.30861724904856560383403885985, −5.53096164188063286123335215275, −5.26994691689554823239706844716, −4.54174436188252590321305161765, −3.77574623727867459499807250738, −3.11999090111784363946957797728, −2.75288179062054437215414622114, −1.94354790731588735069840882354, −1.68071640212912531972121827183, −0.38142142708802669271933037256,
0.38142142708802669271933037256, 1.68071640212912531972121827183, 1.94354790731588735069840882354, 2.75288179062054437215414622114, 3.11999090111784363946957797728, 3.77574623727867459499807250738, 4.54174436188252590321305161765, 5.26994691689554823239706844716, 5.53096164188063286123335215275, 6.30861724904856560383403885985, 6.35826795053767554196357805109, 7.28808761962150163831461415376, 7.33274278909123766119537041455, 7.948282652397653849486575767798, 8.228796414578987652865381425721, 9.034035468909475440393734547129, 9.223980873564047010625797149322, 10.09822296234234106787534183967, 10.21420790210294167924953949616, 10.95434016999755428996981208780
