| L(s) = 1 | + 29·9-s + 30·11-s − 172·19-s + 354·29-s + 424·31-s − 888·41-s − 49·49-s − 288·59-s − 752·61-s − 96·71-s − 178·79-s + 112·81-s − 2.28e3·89-s + 870·99-s − 1.47e3·101-s − 3.32e3·109-s − 1.98e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.10e3·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s + 0.822·11-s − 2.07·19-s + 2.26·29-s + 2.45·31-s − 3.38·41-s − 1/7·49-s − 0.635·59-s − 1.57·61-s − 0.160·71-s − 0.253·79-s + 0.153·81-s − 2.71·89-s + 0.883·99-s − 1.45·101-s − 2.91·109-s − 1.49·121-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 + 1.86·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.392017285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392017285\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4105 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5303 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 86 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21418 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 177 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 p T + p^{3} T^{2} )( 1 + 12 p T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 444 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156898 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14195 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 265354 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 376 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 474790 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 108910 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 89 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 535174 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1140 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1796785 T^{2} + p^{6} T^{4} \) |
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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
−10.40043027654432137687481709923, −9.718402092395656822637018616475, −9.714423249020622395796796655479, −8.702928939529599636692953987768, −8.657949222143302585143803367144, −8.163511437708300661858353916704, −7.83509101081474177901866671296, −6.85836084686403066476002498678, −6.68609321920091941235712463806, −6.56697571482171828761577163184, −5.98459532480821715665335182018, −5.10976797559807628160740589809, −4.72738592974901389271450913809, −4.13507029289574649466371775525, −4.10093768341328458894299391374, −2.96163778967780096077529572159, −2.74848245116596480656520882254, −1.57562108150388869565299711529, −1.49063913908306098977946352513, −0.43885596727286653142658420733,
0.43885596727286653142658420733, 1.49063913908306098977946352513, 1.57562108150388869565299711529, 2.74848245116596480656520882254, 2.96163778967780096077529572159, 4.10093768341328458894299391374, 4.13507029289574649466371775525, 4.72738592974901389271450913809, 5.10976797559807628160740589809, 5.98459532480821715665335182018, 6.56697571482171828761577163184, 6.68609321920091941235712463806, 6.85836084686403066476002498678, 7.83509101081474177901866671296, 8.163511437708300661858353916704, 8.657949222143302585143803367144, 8.702928939529599636692953987768, 9.714423249020622395796796655479, 9.718402092395656822637018616475, 10.40043027654432137687481709923
