| L(s) = 1 | + 108·11-s + 50·19-s − 108·29-s − 542·31-s + 720·41-s + 637·49-s + 252·59-s + 94·61-s + 2.16e3·71-s + 1.13e3·79-s + 2.88e3·89-s − 1.65e3·101-s − 2.55e3·109-s + 6.08e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.36e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.96·11-s + 0.603·19-s − 0.691·29-s − 3.14·31-s + 2.74·41-s + 13/7·49-s + 0.556·59-s + 0.197·61-s + 3.61·71-s + 1.61·79-s + 3.43·89-s − 1.63·101-s − 2.24·109-s + 4.57·121-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 + 0.623·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.823475198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.823475198\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1369 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9502 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 25 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 271 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 132445 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 64838 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296458 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 483877 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1080 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 332882 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 568 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 878510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1440 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1632625 T^{2} + p^{6} T^{4} \) |
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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
−9.492121335054437903208358807526, −9.378385635052092194339047562209, −9.214571536136264920504379289044, −9.082140258860466943296266022024, −8.130800541329968481664752096672, −7.923206977798691821157488472972, −7.17845278542478162134890350893, −7.01019777164308631751311503509, −6.58937134875236234111564721511, −6.06772443958902467981983828011, −5.45849187789021591257034664522, −5.40316328573450888373305853268, −4.25460544231362368151426102862, −4.15261123177848659814572961117, −3.61310313425532991933151187177, −3.31631711913275901123032187752, −2.08055695867881172962423382344, −1.99150276417406948795029389390, −0.978311506715119554673637558918, −0.71826212018856001737328380193,
0.71826212018856001737328380193, 0.978311506715119554673637558918, 1.99150276417406948795029389390, 2.08055695867881172962423382344, 3.31631711913275901123032187752, 3.61310313425532991933151187177, 4.15261123177848659814572961117, 4.25460544231362368151426102862, 5.40316328573450888373305853268, 5.45849187789021591257034664522, 6.06772443958902467981983828011, 6.58937134875236234111564721511, 7.01019777164308631751311503509, 7.17845278542478162134890350893, 7.923206977798691821157488472972, 8.130800541329968481664752096672, 9.082140258860466943296266022024, 9.214571536136264920504379289044, 9.378385635052092194339047562209, 9.492121335054437903208358807526
