| L(s) = 1 | − 16·4-s − 1.19e3·13-s + 256·16-s + 2.20e3·17-s − 2.10e3·23-s + 3.64e3·25-s + 8.20e3·29-s + 1.99e4·43-s + 2.25e4·49-s + 1.91e4·52-s + 1.50e3·53-s − 1.15e5·61-s − 4.09e3·64-s − 3.52e4·68-s + 1.26e5·79-s + 3.36e4·92-s − 5.83e4·100-s + 2.26e5·101-s + 5.00e4·103-s − 4.98e4·107-s − 2.00e5·113-s − 1.31e5·116-s + 3.07e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.96·13-s + 1/4·16-s + 1.84·17-s − 0.827·23-s + 1.16·25-s + 1.81·29-s + 1.64·43-s + 1.34·49-s + 0.981·52-s + 0.0733·53-s − 3.98·61-s − 1/8·64-s − 0.923·68-s + 2.27·79-s + 0.413·92-s − 0.583·100-s + 2.20·101-s + 0.465·103-s − 0.420·107-s − 1.47·113-s − 0.906·116-s + 1.91·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.181678003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.181678003\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 92 p T + p^{5} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3649 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 461 p^{2} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 307702 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 1101 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3583298 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1050 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4104 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35363074 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 62841233 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 141842002 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9995 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 450028765 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 750 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 246071246 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 57920 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2179862870 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 454456379 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 680937230 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 63202 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4802491522 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 189689614 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8570163790 T^{2} + p^{10} T^{4} \) |
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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
−12.01564722989896368056223299065, −10.85941592912127044059202774695, −10.41389127526647543608719727596, −10.23679384610399907726125006126, −9.502633645548166488651127554080, −9.302176830956230348269183314698, −8.662558588919773944038673621331, −7.911105876696710420724617482656, −7.69034022700876328380984609525, −7.20402634972305475145049259457, −6.46717429904714964533298894295, −5.84144785520744976759359268610, −5.31584219160445818996791528294, −4.63755667528941151464541650240, −4.41321701233527808421709477405, −3.33124898745015316029784352709, −2.88290570232400841982222788082, −2.14130213880402589254586535111, −1.07623588320045590845175062375, −0.50840447317880579821769245286,
0.50840447317880579821769245286, 1.07623588320045590845175062375, 2.14130213880402589254586535111, 2.88290570232400841982222788082, 3.33124898745015316029784352709, 4.41321701233527808421709477405, 4.63755667528941151464541650240, 5.31584219160445818996791528294, 5.84144785520744976759359268610, 6.46717429904714964533298894295, 7.20402634972305475145049259457, 7.69034022700876328380984609525, 7.911105876696710420724617482656, 8.662558588919773944038673621331, 9.302176830956230348269183314698, 9.502633645548166488651127554080, 10.23679384610399907726125006126, 10.41389127526647543608719727596, 10.85941592912127044059202774695, 12.01564722989896368056223299065
