| L(s) = 1 | + 50·9-s − 64·11-s + 200·19-s + 100·29-s + 216·31-s + 44·41-s + 650·49-s + 1.00e3·59-s − 1.03e3·61-s − 824·71-s + 1.20e3·79-s + 1.77e3·81-s + 300·89-s − 3.20e3·99-s + 1.40e3·101-s + 1.10e3·109-s + 410·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e3·169-s + ⋯ |
| L(s) = 1 | + 1.85·9-s − 1.75·11-s + 2.41·19-s + 0.640·29-s + 1.25·31-s + 0.167·41-s + 1.89·49-s + 2.20·59-s − 2.17·61-s − 1.37·71-s + 1.70·79-s + 2.42·81-s + 0.357·89-s − 3.24·99-s + 1.38·101-s + 0.966·109-s + 0.308·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.34·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.392595076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.392595076\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9150 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18250 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30550 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 56550 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297750 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 500 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 585650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7150 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 600 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1064050 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1676350 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86475004937360218547711231027, −10.54673404148715868669198788016, −10.08554000731444361154593768878, −9.917023165950861455861547693748, −9.403824827627716724822132422354, −8.824612195946698976107674073743, −8.188848125144498443810314292992, −7.69416537364798965174973992097, −7.31379971720802691878323149035, −7.17941231620303986546456277637, −6.31702851948624255243191570083, −5.79271527161043583096136312479, −4.98085989018928893461698109074, −4.97318672059577961980651840765, −4.19590710006785558926753365408, −3.50158159467581486014116015405, −2.84817102981007986351160784742, −2.23618783113428555914189139512, −1.23295451945082391674409910234, −0.71160351408492586945971911327,
0.71160351408492586945971911327, 1.23295451945082391674409910234, 2.23618783113428555914189139512, 2.84817102981007986351160784742, 3.50158159467581486014116015405, 4.19590710006785558926753365408, 4.97318672059577961980651840765, 4.98085989018928893461698109074, 5.79271527161043583096136312479, 6.31702851948624255243191570083, 7.17941231620303986546456277637, 7.31379971720802691878323149035, 7.69416537364798965174973992097, 8.188848125144498443810314292992, 8.824612195946698976107674073743, 9.403824827627716724822132422354, 9.917023165950861455861547693748, 10.08554000731444361154593768878, 10.54673404148715868669198788016, 10.86475004937360218547711231027
