| L(s) = 1 | − 3·3-s − 50·5-s + 98·7-s − 333·9-s + 601·11-s − 577·13-s + 150·15-s + 41·17-s − 630·19-s − 294·21-s + 442·23-s + 1.87e3·25-s + 1.29e3·27-s + 5.88e3·29-s + 396·31-s − 1.80e3·33-s − 4.90e3·35-s − 8.90e3·37-s + 1.73e3·39-s + 1.77e3·41-s + 2.71e4·43-s + 1.66e4·45-s + 2.12e4·47-s + 7.20e3·49-s − 123·51-s − 5.55e4·53-s − 3.00e4·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.894·5-s + 0.755·7-s − 1.37·9-s + 1.49·11-s − 0.946·13-s + 0.172·15-s + 0.0344·17-s − 0.400·19-s − 0.145·21-s + 0.174·23-s + 3/5·25-s + 0.342·27-s + 1.29·29-s + 0.0740·31-s − 0.288·33-s − 0.676·35-s − 1.06·37-s + 0.182·39-s + 0.164·41-s + 2.23·43-s + 1.22·45-s + 1.40·47-s + 3/7·49-s − 0.00662·51-s − 2.71·53-s − 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 38 p^{2} T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 601 T + 259506 T^{2} - 601 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 577 T + 780172 T^{2} + 577 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 41 T + 1816368 T^{2} - 41 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 630 T + 4931238 T^{2} + 630 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 442 T - 299538 T^{2} - 442 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5885 T + 35168948 T^{2} - 5885 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 396 T + 43423646 T^{2} - 396 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8904 T + 132709718 T^{2} + 8904 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1774 T - 4379094 T^{2} - 1774 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 27122 T + 464103742 T^{2} - 27122 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21289 T + 531685238 T^{2} - 21289 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 55582 T + 1605381282 T^{2} + 55582 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 59600 T + 1881188438 T^{2} + 59600 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 51846 T + 1832119946 T^{2} + 51846 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 45344 T + 2875187158 T^{2} - 45344 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 80744 T + 4781205326 T^{2} + 80744 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13532 T + 3906362902 T^{2} + 13532 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 51795 T + 1771153398 T^{2} - 51795 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 109828 T + 10643414822 T^{2} + 109828 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 37650 T + 10990453658 T^{2} + 37650 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 96339 T + 16335863448 T^{2} + 96339 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.529165607519661623166775589020, −9.309386647600035620752630791638, −8.687553717694584009522987546787, −8.662058247548139649540697740061, −7.81101878079222980527451182398, −7.71797098091353207613355295066, −7.11144057133094406540850736891, −6.56202364446871204345681226064, −6.06890780441882060398775229496, −5.73436923292498115752312686484, −4.84322583691820073851481188193, −4.69754928714166992728916764065, −4.12525016876140056879242745210, −3.55654830105276651465942001897, −2.80063126979602038027110855925, −2.56193815301449967133645433396, −1.45799336295924251818947666775, −1.13292691599672949504937901900, 0, 0,
1.13292691599672949504937901900, 1.45799336295924251818947666775, 2.56193815301449967133645433396, 2.80063126979602038027110855925, 3.55654830105276651465942001897, 4.12525016876140056879242745210, 4.69754928714166992728916764065, 4.84322583691820073851481188193, 5.73436923292498115752312686484, 6.06890780441882060398775229496, 6.56202364446871204345681226064, 7.11144057133094406540850736891, 7.71797098091353207613355295066, 7.81101878079222980527451182398, 8.662058247548139649540697740061, 8.687553717694584009522987546787, 9.309386647600035620752630791638, 9.529165607519661623166775589020
