| L(s) = 1 | + 10·5-s − 34·9-s + 124·13-s − 92·17-s + 75·25-s + 180·29-s + 428·37-s − 20·41-s − 340·45-s + 294·49-s + 1.35e3·53-s − 500·61-s + 1.24e3·65-s + 1.04e3·73-s + 427·81-s − 920·85-s + 1.94e3·89-s − 1.86e3·97-s + 1.20e3·101-s − 4.30e3·109-s − 4.36e3·113-s − 4.21e3·117-s − 2.58e3·121-s + 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.25·9-s + 2.64·13-s − 1.31·17-s + 3/5·25-s + 1.15·29-s + 1.90·37-s − 0.0761·41-s − 1.12·45-s + 6/7·49-s + 3.51·53-s − 1.04·61-s + 2.36·65-s + 1.67·73-s + 0.585·81-s − 1.17·85-s + 2.31·89-s − 1.95·97-s + 1.18·101-s − 3.78·109-s − 3.63·113-s − 3.33·117-s − 1.93·121-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
| \(L(2)\) |
\(\approx\) |
\(3.243707169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.243707169\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 34 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2582 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2198 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12646 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 36462 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 154514 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 49226 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 241478 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 599106 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 581342 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 522 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 217758 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 999074 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 970 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 934 T + p^{3} T^{2} )^{2} \) |
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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
−11.28904346255545418682715934292, −10.97748638155270677193538425627, −10.55286502382544894158109623122, −10.26262158657310536041491353365, −9.261281330206464793554102825460, −9.121500481511738986138752906983, −8.699027344427225797490265840023, −8.248677103741283532669548678471, −7.83709211555687191639217150466, −6.82003109683543425815133556034, −6.35009909812789073548131244174, −6.22585159214282625388433696280, −5.52194766478119576004314393366, −5.16826921682130328249364668209, −3.99666622616855055289384092020, −3.95065408755848868917436086590, −2.72439308021699582370502183569, −2.52041268001832373920895730493, −1.39120333416588061945260567780, −0.70740526746535172443980076197,
0.70740526746535172443980076197, 1.39120333416588061945260567780, 2.52041268001832373920895730493, 2.72439308021699582370502183569, 3.95065408755848868917436086590, 3.99666622616855055289384092020, 5.16826921682130328249364668209, 5.52194766478119576004314393366, 6.22585159214282625388433696280, 6.35009909812789073548131244174, 6.82003109683543425815133556034, 7.83709211555687191639217150466, 8.248677103741283532669548678471, 8.699027344427225797490265840023, 9.121500481511738986138752906983, 9.261281330206464793554102825460, 10.26262158657310536041491353365, 10.55286502382544894158109623122, 10.97748638155270677193538425627, 11.28904346255545418682715934292
