| L(s) = 1 | − 8·7-s − 10·9-s + 132·17-s + 264·23-s − 25·25-s − 304·31-s + 876·41-s + 408·47-s − 638·49-s + 80·63-s + 864·71-s − 724·73-s + 320·79-s − 629·81-s − 1.62e3·89-s + 2.21e3·97-s − 2.07e3·113-s − 1.05e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.32e3·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 0.370·9-s + 1.88·17-s + 2.39·23-s − 1/5·25-s − 1.76·31-s + 3.33·41-s + 1.26·47-s − 1.86·49-s + 0.159·63-s + 1.44·71-s − 1.16·73-s + 0.455·79-s − 0.862·81-s − 1.92·89-s + 2.31·97-s − 1.72·113-s − 0.813·119-s + 1.89·121-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 − 0.697·153-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.951720314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951720314\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40678 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 157990 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 248470 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 234358 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 359642 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 447050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 362 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1138390 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{2} \) |
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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
−9.717141575532191684185157017954, −9.001991498319209293192737163879, −8.975287280805214976339080733048, −8.142916655522256424869248973162, −7.935646030344014569655680072488, −7.32413559426443359812558459997, −7.20253056205684990320231509499, −6.72193077318100427533334263103, −5.96319200181216474610487650586, −5.77795850273166173665972315547, −5.43672880766219590606009387896, −4.86506294395467970151436499211, −4.41180159246136435423391767744, −3.65300097773270333920247806650, −3.45617477976136811041237087102, −2.79035127196225421676899351848, −2.52163419297756270846207188530, −1.52827403572224603466696020579, −1.04214109859367355837513261597, −0.46513673086794340581250285551,
0.46513673086794340581250285551, 1.04214109859367355837513261597, 1.52827403572224603466696020579, 2.52163419297756270846207188530, 2.79035127196225421676899351848, 3.45617477976136811041237087102, 3.65300097773270333920247806650, 4.41180159246136435423391767744, 4.86506294395467970151436499211, 5.43672880766219590606009387896, 5.77795850273166173665972315547, 5.96319200181216474610487650586, 6.72193077318100427533334263103, 7.20253056205684990320231509499, 7.32413559426443359812558459997, 7.935646030344014569655680072488, 8.142916655522256424869248973162, 8.975287280805214976339080733048, 9.001991498319209293192737163879, 9.717141575532191684185157017954
