| L(s) = 1 | − 4·4-s + 38·9-s + 120·11-s + 16·16-s + 104·19-s + 468·29-s − 608·31-s − 152·36-s − 108·41-s − 480·44-s − 49·49-s + 888·59-s + 76·61-s − 64·64-s − 1.44e3·71-s − 416·76-s + 1.61e3·79-s + 715·81-s − 2.29e3·89-s + 4.56e3·99-s + 3.90e3·101-s − 1.51e3·109-s − 1.87e3·116-s + 8.13e3·121-s + 2.43e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.40·9-s + 3.28·11-s + 1/4·16-s + 1.25·19-s + 2.99·29-s − 3.52·31-s − 0.703·36-s − 0.411·41-s − 1.64·44-s − 1/7·49-s + 1.95·59-s + 0.159·61-s − 1/8·64-s − 2.40·71-s − 0.627·76-s + 2.30·79-s + 0.980·81-s − 2.72·89-s + 4.62·99-s + 3.84·101-s − 1.33·109-s − 1.49·116-s + 6.11·121-s + 1.76·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.023266513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.023266513\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8062 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 304 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 90070 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 120598 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 94750 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105910 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 444 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 38 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 374618 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 p^{2} T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 808 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1146 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1820446 T^{2} + p^{6} T^{4} \) |
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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
−11.37643093717761777502401391178, −10.86978330661889644476079503747, −10.14727476139620376238470723790, −9.882872038653941858531887170284, −9.409198073231484829188239194126, −8.966391104512517363372546779399, −8.797905911506400684017681057288, −8.041363228008842097029675423455, −7.28818648413645231568892132242, −6.88746191593390064988391444840, −6.75855832414526605873794208356, −5.98280822643636873047795661413, −5.41142698384425784560997634317, −4.54507723417369765018060943621, −4.32157689798628243884111798649, −3.61214636447142368286377655970, −3.36711450285520587675833666168, −1.91893742744381110289285519358, −1.30772850384035401753797417953, −0.860167930939699129271853885946,
0.860167930939699129271853885946, 1.30772850384035401753797417953, 1.91893742744381110289285519358, 3.36711450285520587675833666168, 3.61214636447142368286377655970, 4.32157689798628243884111798649, 4.54507723417369765018060943621, 5.41142698384425784560997634317, 5.98280822643636873047795661413, 6.75855832414526605873794208356, 6.88746191593390064988391444840, 7.28818648413645231568892132242, 8.041363228008842097029675423455, 8.797905911506400684017681057288, 8.966391104512517363372546779399, 9.409198073231484829188239194126, 9.882872038653941858531887170284, 10.14727476139620376238470723790, 10.86978330661889644476079503747, 11.37643093717761777502401391178
