| L(s) = 1 | − 8·4-s + 26·7-s − 178·13-s + 64·16-s + 202·19-s − 208·28-s − 182·31-s + 4.49e3·37-s − 250·43-s − 4.29e3·49-s + 1.42e3·52-s − 9.95e3·61-s − 512·64-s + 7.65e3·67-s − 4.14e3·73-s − 1.61e3·76-s − 2.46e4·79-s − 4.62e3·91-s + 2.15e4·97-s − 2.24e4·103-s − 3.64e4·109-s + 1.66e3·112-s + 2.27e4·121-s + 1.45e3·124-s + 127-s + 131-s + 5.25e3·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.530·7-s − 1.05·13-s + 1/4·16-s + 0.559·19-s − 0.265·28-s − 0.189·31-s + 3.28·37-s − 0.135·43-s − 1.78·49-s + 0.526·52-s − 2.67·61-s − 1/8·64-s + 1.70·67-s − 0.777·73-s − 0.279·76-s − 3.94·79-s − 0.558·91-s + 2.29·97-s − 2.12·103-s − 3.07·109-s + 0.132·112-s + 1.55·121-s + 0.0946·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 0.296·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.494612201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494612201\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 13 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 22784 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 89 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 165584 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 101 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 287360 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 947984 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 91 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2248 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5409314 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 125 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7228112 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10686530 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8230960 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4975 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3829 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 42203810 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2072 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12304 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 16366064 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 123547970 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10783 T + p^{4} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83098604464351309208188895800, −10.05664628097467282997382419853, −9.740855766387359119940689346922, −9.499732954945772953201488717529, −8.958485834367537736792662894532, −8.366987542013610854563930122431, −7.968405304692802912502350950340, −7.50307687889942358203122218475, −7.25459386716443578845856551977, −6.32230060229842747822549693667, −6.11691036340097980797332403878, −5.29170357531017789407645999460, −5.02387734069161994127647736574, −4.24779456690378979188378306747, −4.20064241339002762369128715685, −2.92602668684943476428613873929, −2.87529584017056264041298525230, −1.82007605958556108608993297326, −1.20437254670244510996478023714, −0.35663300313941110676378234481,
0.35663300313941110676378234481, 1.20437254670244510996478023714, 1.82007605958556108608993297326, 2.87529584017056264041298525230, 2.92602668684943476428613873929, 4.20064241339002762369128715685, 4.24779456690378979188378306747, 5.02387734069161994127647736574, 5.29170357531017789407645999460, 6.11691036340097980797332403878, 6.32230060229842747822549693667, 7.25459386716443578845856551977, 7.50307687889942358203122218475, 7.968405304692802912502350950340, 8.366987542013610854563930122431, 8.958485834367537736792662894532, 9.499732954945772953201488717529, 9.740855766387359119940689346922, 10.05664628097467282997382419853, 10.83098604464351309208188895800
