| L(s) = 1 | + 8·2-s + 32·4-s + 76·5-s + 608·10-s + 1.43e3·13-s − 1.02e3·16-s − 1.43e3·17-s + 2.43e3·20-s + 2.65e3·25-s + 1.15e4·26-s − 8.19e3·32-s − 1.14e4·34-s − 2.35e4·37-s + 9.90e3·41-s + 2.12e4·50-s + 4.60e4·52-s + 4.75e4·53-s − 1.09e5·61-s − 3.27e4·64-s + 1.09e5·65-s − 4.58e4·68-s − 6.86e4·73-s − 1.88e5·74-s − 7.78e4·80-s − 5.90e4·81-s + 7.92e4·82-s − 1.08e5·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.35·5-s + 1.92·10-s + 2.35·13-s − 16-s − 1.20·17-s + 1.35·20-s + 0.848·25-s + 3.33·26-s − 1.41·32-s − 1.70·34-s − 2.82·37-s + 0.920·41-s + 1.19·50-s + 2.35·52-s + 2.32·53-s − 3.78·61-s − 64-s + 3.20·65-s − 1.20·68-s − 1.50·73-s − 3.99·74-s − 1.35·80-s − 81-s + 1.30·82-s − 1.63·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.218326603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.218326603\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{5} T^{2} \) |
| 5 | $C_2$ | \( 1 - 76 T + p^{5} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 1194 T + p^{5} T^{2} )( 1 - 244 T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 808 T + p^{5} T^{2} )( 1 + 2242 T + p^{5} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2950 T + p^{5} T^{2} )( 1 + 2950 T + p^{5} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11292 T + p^{5} T^{2} )( 1 + 12242 T + p^{5} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4952 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 40244 T + p^{5} T^{2} )( 1 - 7294 T + p^{5} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 54948 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 20144 T + p^{5} T^{2} )( 1 + 88806 T + p^{5} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 51050 T + p^{5} T^{2} )( 1 + 51050 T + p^{5} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 160808 T + p^{5} T^{2} )( 1 + 92142 T + p^{5} T^{2} ) \) |
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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
−17.68042566346976468088043014887, −16.94057795811067793899836132142, −15.97153468610770875600758445752, −15.64122081029977962749113580978, −14.92479841635355311450675549347, −13.82992199286038374655354623587, −13.80684094386807822589105846901, −13.30080491769444111585650901095, −12.60433177603834056131724293569, −11.71503586664072084945421664084, −10.89269666887750239271734571355, −10.35347358724105618611397896637, −8.928426237167835127411821879286, −8.817869440260561836869181617063, −6.98708908057612980006958061687, −6.09864020149309213865293067008, −5.71670857638454933790124513986, −4.46409222812172831304990880757, −3.31923024199394702806445294093, −1.82464575287869434280692369639,
1.82464575287869434280692369639, 3.31923024199394702806445294093, 4.46409222812172831304990880757, 5.71670857638454933790124513986, 6.09864020149309213865293067008, 6.98708908057612980006958061687, 8.817869440260561836869181617063, 8.928426237167835127411821879286, 10.35347358724105618611397896637, 10.89269666887750239271734571355, 11.71503586664072084945421664084, 12.60433177603834056131724293569, 13.30080491769444111585650901095, 13.80684094386807822589105846901, 13.82992199286038374655354623587, 14.92479841635355311450675549347, 15.64122081029977962749113580978, 15.97153468610770875600758445752, 16.94057795811067793899836132142, 17.68042566346976468088043014887
