| L(s) = 1 | − 5·3-s + 5·5-s + 27·9-s + 15·11-s + 26·13-s − 25·15-s − 27·17-s − 154·19-s + 186·23-s − 280·27-s + 6·29-s − 328·31-s − 75·33-s − 254·37-s − 130·39-s − 192·41-s + 268·43-s + 135·45-s + 51·47-s + 135·51-s − 240·53-s + 75·55-s + 770·57-s − 396·59-s − 616·61-s + 130·65-s − 296·67-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.447·5-s + 9-s + 0.411·11-s + 0.554·13-s − 0.430·15-s − 0.385·17-s − 1.85·19-s + 1.68·23-s − 1.99·27-s + 0.0384·29-s − 1.90·31-s − 0.395·33-s − 1.12·37-s − 0.533·39-s − 0.731·41-s + 0.950·43-s + 0.447·45-s + 0.158·47-s + 0.370·51-s − 0.622·53-s + 0.183·55-s + 1.78·57-s − 0.873·59-s − 1.29·61-s + 0.248·65-s − 0.539·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1067709361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1067709361\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 27 T - 4184 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 154 T + 16857 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 186 T + 22429 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 328 T + 77793 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 134 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 51 T - 101222 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 240 T - 91277 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 396 T - 48563 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 616 T + 152475 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 296 T - 213147 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 322 T - 285333 T^{2} + 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 659 T - 58758 T^{2} + 659 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 300 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1020 T + 335431 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 199 T + p^{3} T^{2} )^{2} \) |
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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
−10.32323225063286106676204367390, −9.255109555070225102644864004464, −9.045421544376108222864052123903, −8.815583880703080147166424819927, −8.207718243924617072248016641471, −7.47258036429880613369264215433, −7.31672064278767358603216596795, −6.72995060631504894236144495922, −6.43141615348541834869962123388, −5.90103358176555698201045829837, −5.69003541410506476139286667585, −4.96496748883927920799702953109, −4.72546423493906028613694773667, −3.99927793883202316040319704788, −3.75431605309231219861817224812, −2.99959866714197180010797992242, −2.20627018580590331258230655295, −1.61232492289303765167476126310, −1.27327542420351744444946177677, −0.088561251189226392705214232137,
0.088561251189226392705214232137, 1.27327542420351744444946177677, 1.61232492289303765167476126310, 2.20627018580590331258230655295, 2.99959866714197180010797992242, 3.75431605309231219861817224812, 3.99927793883202316040319704788, 4.72546423493906028613694773667, 4.96496748883927920799702953109, 5.69003541410506476139286667585, 5.90103358176555698201045829837, 6.43141615348541834869962123388, 6.72995060631504894236144495922, 7.31672064278767358603216596795, 7.47258036429880613369264215433, 8.207718243924617072248016641471, 8.815583880703080147166424819927, 9.045421544376108222864052123903, 9.255109555070225102644864004464, 10.32323225063286106676204367390
