| L(s) = 1 | − 9·9-s − 112·11-s − 8·19-s + 412·29-s − 304·31-s − 492·41-s + 286·49-s − 112·59-s − 4·61-s − 1.34e3·71-s − 816·79-s + 81·81-s − 132·89-s + 1.00e3·99-s − 396·101-s − 124·109-s + 6.74e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.00e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 3.06·11-s − 0.0965·19-s + 2.63·29-s − 1.76·31-s − 1.87·41-s + 0.833·49-s − 0.247·59-s − 0.00839·61-s − 2.24·71-s − 1.16·79-s + 1/9·81-s − 0.157·89-s + 1.02·99-s − 0.390·101-s − 0.108·109-s + 5.06·121-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 − 1.36·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3002 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5838 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 21782 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10730 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 206046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 281878 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 450982 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 672 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 590866 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 697350 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 967870 T^{2} + p^{6} T^{4} \) |
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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
−10.02625051401963210467891274654, −9.961330259366870301714752079040, −9.034437486085000684684456868724, −8.522105176246537065099730048452, −8.420965240853881810552966229208, −7.86197692270884753898154690220, −7.27567543719548659570757669931, −7.21287775648466853315084680363, −6.28594904743092998073887404083, −5.93299351961544439853575582028, −5.20234415738391452486477451965, −5.12500783878217281851563795407, −4.59538274183147697469467042266, −3.77651573908943329558227462050, −2.99618877926623599753681107198, −2.74502401999040569148335862030, −2.17341438147151103132788865928, −1.23501937385760517758318041152, 0, 0,
1.23501937385760517758318041152, 2.17341438147151103132788865928, 2.74502401999040569148335862030, 2.99618877926623599753681107198, 3.77651573908943329558227462050, 4.59538274183147697469467042266, 5.12500783878217281851563795407, 5.20234415738391452486477451965, 5.93299351961544439853575582028, 6.28594904743092998073887404083, 7.21287775648466853315084680363, 7.27567543719548659570757669931, 7.86197692270884753898154690220, 8.420965240853881810552966229208, 8.522105176246537065099730048452, 9.034437486085000684684456868724, 9.961330259366870301714752079040, 10.02625051401963210467891274654
