| L(s) = 1 | + 10·5-s − 2·9-s − 68·13-s + 228·17-s + 75·25-s + 52·29-s + 300·37-s + 684·41-s − 20·45-s − 634·49-s + 524·53-s + 524·61-s − 680·65-s + 1.36e3·73-s − 725·81-s + 2.28e3·85-s − 1.26e3·89-s − 1.93e3·97-s − 3.27e3·101-s + 684·109-s + 4.21e3·113-s + 136·117-s − 790·121-s + 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.0740·9-s − 1.45·13-s + 3.25·17-s + 3/5·25-s + 0.332·29-s + 1.33·37-s + 2.60·41-s − 0.0662·45-s − 1.84·49-s + 1.35·53-s + 1.09·61-s − 1.29·65-s + 2.18·73-s − 0.994·81-s + 2.90·85-s − 1.50·89-s − 2.02·97-s − 3.22·101-s + 0.601·109-s + 3.50·113-s + 0.107·117-s − 0.593·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.487758217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.487758217\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 634 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 790 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19398 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 49390 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 47374 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 133526 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 262 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 170310 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 262 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 353954 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 392610 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 682 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 945310 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1120642 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 966 T + p^{3} T^{2} )^{2} \) |
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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
−11.18979244484629703069260226718, −11.17232895040403423161633032483, −10.30353078860145053682375987519, −9.788726723249713355415038476614, −9.694558696368641783800383550835, −9.521872490500277563366716824698, −8.470893821883778292385435232823, −8.137558106395332116037161490464, −7.48682571610062326990080666954, −7.32121019327742009457768664269, −6.51600209933778470726675759750, −5.86084239021956243696853635145, −5.53924692072056636305355400194, −5.11170736274355173469758393035, −4.37927254744595225945716410482, −3.62460652763853189192581789228, −2.81491063073298123861461612906, −2.47548262652992142409251432209, −1.36986175199117627351380614427, −0.74541862770075387002929659494,
0.74541862770075387002929659494, 1.36986175199117627351380614427, 2.47548262652992142409251432209, 2.81491063073298123861461612906, 3.62460652763853189192581789228, 4.37927254744595225945716410482, 5.11170736274355173469758393035, 5.53924692072056636305355400194, 5.86084239021956243696853635145, 6.51600209933778470726675759750, 7.32121019327742009457768664269, 7.48682571610062326990080666954, 8.137558106395332116037161490464, 8.470893821883778292385435232823, 9.521872490500277563366716824698, 9.694558696368641783800383550835, 9.788726723249713355415038476614, 10.30353078860145053682375987519, 11.17232895040403423161633032483, 11.18979244484629703069260226718
