| L(s) = 1 | − 26·4-s + 38·5-s − 348·16-s − 988·20-s − 6.14e3·23-s − 5.16e3·25-s − 3.16e3·31-s − 1.82e4·37-s + 3.32e4·47-s − 3.11e4·49-s + 3.25e4·53-s − 2.90e4·59-s + 3.56e4·64-s − 2.12e4·67-s − 6.20e4·71-s − 1.32e4·80-s + 2.18e5·89-s + 1.59e5·92-s − 2.72e4·97-s + 1.34e5·100-s + 3.59e5·103-s + 1.76e5·113-s − 2.33e5·115-s + 8.22e4·124-s − 3.28e5·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.812·4-s + 0.679·5-s − 0.339·16-s − 0.552·20-s − 2.42·23-s − 1.65·25-s − 0.590·31-s − 2.19·37-s + 2.19·47-s − 1.85·49-s + 1.59·53-s − 1.08·59-s + 1.08·64-s − 0.578·67-s − 1.46·71-s − 0.231·80-s + 2.93·89-s + 1.96·92-s − 0.293·97-s + 1.34·100-s + 3.33·103-s + 1.29·113-s − 1.64·115-s + 0.480·124-s − 1.88·125-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06674717275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06674717275\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 13 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 19 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 31182 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 195386 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 489694 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 23970 p T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3071 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40779098 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 51 p T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9145 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 71336686 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 95089014 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 16636 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 16266 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 14505 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 1628480154 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10635 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 31045 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3011641938 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 857839330 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 683252486 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 109481 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13615 T + p^{5} T^{2} )^{2} \) |
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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
−9.218914602035423218190824574565, −9.031365235729130182007577972943, −8.601290677012929719194727650378, −8.230981878997579554429563816155, −7.57726041344251753854507321027, −7.46078585879238718949522874821, −6.87497083406518797554126573233, −6.13093197836481411263017348054, −5.94872918095503303749004367801, −5.71815910880339487104931762360, −4.91541441667594302836423836608, −4.74293044797740856655873966393, −3.99515702880876473292315775602, −3.78043299515678555449774760790, −3.29077468338583731975157725145, −2.37057992129173205865520783318, −1.92418412793074398321649732816, −1.77487286863410533771084629080, −0.796517983491213191280690291787, −0.05853081948169259370172578000,
0.05853081948169259370172578000, 0.796517983491213191280690291787, 1.77487286863410533771084629080, 1.92418412793074398321649732816, 2.37057992129173205865520783318, 3.29077468338583731975157725145, 3.78043299515678555449774760790, 3.99515702880876473292315775602, 4.74293044797740856655873966393, 4.91541441667594302836423836608, 5.71815910880339487104931762360, 5.94872918095503303749004367801, 6.13093197836481411263017348054, 6.87497083406518797554126573233, 7.46078585879238718949522874821, 7.57726041344251753854507321027, 8.230981878997579554429563816155, 8.601290677012929719194727650378, 9.031365235729130182007577972943, 9.218914602035423218190824574565
