| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1825830923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1825830923\) |
| \(L(3)\) |
|
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
−10.93631210198697436782615392123, −10.16199491940091597242675799113, −9.891322574879371676670946406868, −9.207045733646578781260160022598, −8.999759047370942090470917468934, −8.453008034887991729264471645022, −8.157343070768117637287995643285, −7.32319757431566479749559775770, −7.16190900322369066839422807967, −6.40106797764017912637946688636, −6.06081140584758020632294284492, −5.46512174917530169104477686388, −5.00462930779560786770582878062, −4.37698825252989870317821925238, −3.77037158751806346886892945813, −3.24223649320747019104340451182, −2.86233150434313959491818759232, −1.55522247558368352398877429537, −1.43673730378897259929716533133, −0.11773382768510361435261314487,
0.11773382768510361435261314487, 1.43673730378897259929716533133, 1.55522247558368352398877429537, 2.86233150434313959491818759232, 3.24223649320747019104340451182, 3.77037158751806346886892945813, 4.37698825252989870317821925238, 5.00462930779560786770582878062, 5.46512174917530169104477686388, 6.06081140584758020632294284492, 6.40106797764017912637946688636, 7.16190900322369066839422807967, 7.32319757431566479749559775770, 8.157343070768117637287995643285, 8.453008034887991729264471645022, 8.999759047370942090470917468934, 9.207045733646578781260160022598, 9.891322574879371676670946406868, 10.16199491940091597242675799113, 10.93631210198697436782615392123
