L(s) = 1 | − 4·4-s − 60·11-s + 16·16-s + 98·19-s + 372·29-s − 320·31-s + 756·41-s + 240·44-s + 517·49-s − 408·59-s − 1.75e3·61-s − 64·64-s − 1.21e3·71-s − 392·76-s − 2.30e3·79-s − 1.96e3·89-s − 2.49e3·101-s − 4.39e3·109-s − 1.48e3·116-s + 38·121-s + 1.28e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1.64·11-s + 1/4·16-s + 1.18·19-s + 2.38·29-s − 1.85·31-s + 2.87·41-s + 0.822·44-s + 1.50·49-s − 0.900·59-s − 3.68·61-s − 1/8·64-s − 2.02·71-s − 0.591·76-s − 3.27·79-s − 2.34·89-s − 2.45·101-s − 3.86·109-s − 1.19·116-s + 0.0285·121-s + 0.926·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.03419399126\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03419399126\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 517 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 673 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9682 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 49 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 186 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 93025 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 87190 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 186910 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 27146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 877 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566557 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 606 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 592273 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1151 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1133170 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 984 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1755121 T^{2} + p^{6} T^{4} \) |
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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
−9.564761769758543862653475748844, −8.953832419026667716413265868582, −8.790808247148220346110217456488, −8.007365240798604708540257608186, −7.970677178766323666608083303722, −7.35681374912598594798359376045, −7.26340383987558196194507986400, −6.60371642993232618353889771195, −5.81714220743060773863603325223, −5.73182144320499437276733258103, −5.39446144541566994538082280386, −4.71352126708001978741484292084, −4.34518000464995601005167117026, −4.06194061174577514478596781255, −3.06568724449418138313411360358, −2.82723593440276074018956128534, −2.57867646435804228174035187787, −1.40439505451726550675672085872, −1.17967776171013015864471717909, −0.04564443256450501203090249758,
0.04564443256450501203090249758, 1.17967776171013015864471717909, 1.40439505451726550675672085872, 2.57867646435804228174035187787, 2.82723593440276074018956128534, 3.06568724449418138313411360358, 4.06194061174577514478596781255, 4.34518000464995601005167117026, 4.71352126708001978741484292084, 5.39446144541566994538082280386, 5.73182144320499437276733258103, 5.81714220743060773863603325223, 6.60371642993232618353889771195, 7.26340383987558196194507986400, 7.35681374912598594798359376045, 7.970677178766323666608083303722, 8.007365240798604708540257608186, 8.790808247148220346110217456488, 8.953832419026667716413265868582, 9.564761769758543862653475748844