| L(s) = 1 | + 38·9-s + 88·11-s + 88·19-s − 396·29-s + 320·31-s − 396·41-s + 110·49-s − 1.33e3·59-s + 1.10e3·61-s − 1.45e3·71-s − 1.31e3·79-s + 715·81-s − 1.42e3·89-s + 3.34e3·99-s + 3.13e3·101-s + 3.98e3·109-s + 3.14e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.91e3·169-s + ⋯ |
| L(s) = 1 | + 1.40·9-s + 2.41·11-s + 1.06·19-s − 2.53·29-s + 1.85·31-s − 1.50·41-s + 0.320·49-s − 2.94·59-s + 2.30·61-s − 2.43·71-s − 1.86·79-s + 0.980·81-s − 1.70·89-s + 3.39·99-s + 3.08·101-s + 3.50·109-s + 2.36·121-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 + 1.77·169-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
| \(L(2)\) |
\(\approx\) |
\(3.900563505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.900563505\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7326 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21198 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75062 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 71138 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 239190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 668 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 550 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 728 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 754318 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 656 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1087878 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 714 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1596862 T^{2} + p^{6} T^{4} \) |
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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
−11.37062148004353376666892853974, −10.51144710145822747216986435357, −9.967084439252072432664382929900, −9.793344234202048404217439665338, −9.359780626147063936784412224771, −8.755846155833451657849190727595, −8.579479293274781950328109718903, −7.51944621311821736929631191100, −7.38257454321336002686974026754, −6.95461962246795676402372836122, −6.30576197466681757589938194497, −6.01079322619694650630614098673, −5.25154949180202388889828133407, −4.43799956364972988386813805779, −4.26910800941007446495393960321, −3.56321133967256004305705441496, −3.10486878248920162524633153405, −1.71832803453461965480116944007, −1.58275458047137451576822410884, −0.71747529165514831435238366391,
0.71747529165514831435238366391, 1.58275458047137451576822410884, 1.71832803453461965480116944007, 3.10486878248920162524633153405, 3.56321133967256004305705441496, 4.26910800941007446495393960321, 4.43799956364972988386813805779, 5.25154949180202388889828133407, 6.01079322619694650630614098673, 6.30576197466681757589938194497, 6.95461962246795676402372836122, 7.38257454321336002686974026754, 7.51944621311821736929631191100, 8.579479293274781950328109718903, 8.755846155833451657849190727595, 9.359780626147063936784412224771, 9.793344234202048404217439665338, 9.967084439252072432664382929900, 10.51144710145822747216986435357, 11.37062148004353376666892853974
