| L(s) = 1 | − 90·5-s + 90·9-s − 504·11-s + 440·19-s + 4.97e3·25-s − 1.38e4·29-s − 1.35e4·31-s − 396·41-s − 8.10e3·45-s + 3.00e4·49-s + 4.53e4·55-s + 4.93e4·59-s − 1.13e4·61-s − 1.06e5·71-s − 1.03e5·79-s − 5.09e4·81-s − 1.99e4·89-s − 3.96e4·95-s − 4.53e4·99-s − 2.18e5·101-s − 4.20e4·109-s − 1.31e5·121-s − 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.60·5-s + 0.370·9-s − 1.25·11-s + 0.279·19-s + 1.59·25-s − 3.06·29-s − 2.52·31-s − 0.0367·41-s − 0.596·45-s + 1.78·49-s + 2.02·55-s + 1.84·59-s − 0.392·61-s − 2.51·71-s − 1.87·79-s − 0.862·81-s − 0.267·89-s − 0.450·95-s − 0.465·99-s − 2.12·101-s − 0.338·109-s − 0.817·121-s − 0.953·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3442213342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3442213342\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 18 p T + p^{5} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 30050 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 252 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 728330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2363810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6946370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56462470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293842250 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 347593490 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 802472090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 24660 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 795787610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 883886830 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4053674810 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6923133890 T^{2} + p^{10} T^{4} \) |
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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
−14.49752594411039763082254060543, −12.90983674890392720471875134896, −12.85937452426659599286633648871, −11.90777150247344623859242650377, −11.58005729088650790866170375174, −10.81171708991485132610084415289, −10.73355844752777835638069157217, −9.770690845478940023450445049360, −9.095376398425732577586925992651, −8.540671969408024928511524549821, −7.75938724834058794437686308689, −7.25338803591288837699092310252, −7.20400959014858528732836540970, −5.58315848127763899319032472826, −5.43403235251834930132038685870, −4.16887342677639859435482830729, −3.83913336687543553481393853115, −2.90215463686548024180531290082, −1.72972365910079842681397333888, −0.24545829951886585968100040249,
0.24545829951886585968100040249, 1.72972365910079842681397333888, 2.90215463686548024180531290082, 3.83913336687543553481393853115, 4.16887342677639859435482830729, 5.43403235251834930132038685870, 5.58315848127763899319032472826, 7.20400959014858528732836540970, 7.25338803591288837699092310252, 7.75938724834058794437686308689, 8.540671969408024928511524549821, 9.095376398425732577586925992651, 9.770690845478940023450445049360, 10.73355844752777835638069157217, 10.81171708991485132610084415289, 11.58005729088650790866170375174, 11.90777150247344623859242650377, 12.85937452426659599286633648871, 12.90983674890392720471875134896, 14.49752594411039763082254060543
