| L(s) = 1 | − 8·4-s − 166·7-s − 82·13-s + 64·16-s − 278·19-s + 1.32e3·28-s − 2.10e3·31-s + 3.34e3·37-s + 5.03e3·43-s + 1.58e4·49-s + 656·52-s + 1.16e4·61-s − 512·64-s − 1.45e4·67-s + 9.10e3·73-s + 2.22e3·76-s + 1.85e4·79-s + 1.36e4·91-s − 3.58e3·97-s + 3.74e4·103-s + 3.02e4·109-s − 1.06e4·112-s + 2.27e4·121-s + 1.68e4·124-s + 127-s + 131-s + 4.61e4·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.38·7-s − 0.485·13-s + 1/4·16-s − 0.770·19-s + 1.69·28-s − 2.18·31-s + 2.44·37-s + 2.72·43-s + 6.60·49-s + 0.242·52-s + 3.13·61-s − 1/8·64-s − 3.23·67-s + 1.70·73-s + 0.385·76-s + 2.97·79-s + 1.64·91-s − 0.381·97-s + 3.52·103-s + 2.54·109-s − 0.846·112-s + 1.55·121-s + 1.09·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 2.60·133-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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.158694494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158694494\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + 83 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 22784 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 41 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 96496 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 139 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 509120 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 959504 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 1051 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1672 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4960034 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2515 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1064912 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15628610 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6800 p^{2} T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5825 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7259 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3116450 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4552 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9296 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 31229744 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 88573250 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1793 T + p^{4} T^{2} )^{2} \) |
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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
−10.44046630923034849601665517538, −10.21785883527478271901452652081, −9.586548407013225300273707336271, −9.515893346906762112575587115335, −8.964516087520742296047036227631, −8.857858904524668079703929288711, −7.77999410508680395803883763916, −7.44434584397889287308884776917, −6.97709441275613173985812448423, −6.39496174273941335776903917652, −5.99295224600861205571582951398, −5.83644049865966128632431288171, −4.95935506548344604362020961435, −4.17007130197752885590556317704, −3.71982074317332032159628536422, −3.39008991785455277909617283627, −2.59181272150738633398951179649, −2.27698905395963172246267944663, −0.61683659236779295392200735955, −0.51653426168262493240329966393,
0.51653426168262493240329966393, 0.61683659236779295392200735955, 2.27698905395963172246267944663, 2.59181272150738633398951179649, 3.39008991785455277909617283627, 3.71982074317332032159628536422, 4.17007130197752885590556317704, 4.95935506548344604362020961435, 5.83644049865966128632431288171, 5.99295224600861205571582951398, 6.39496174273941335776903917652, 6.97709441275613173985812448423, 7.44434584397889287308884776917, 7.77999410508680395803883763916, 8.857858904524668079703929288711, 8.964516087520742296047036227631, 9.515893346906762112575587115335, 9.586548407013225300273707336271, 10.21785883527478271901452652081, 10.44046630923034849601665517538
