| L(s) = 1 | − 4·5-s + 140·11-s + 48·19-s − 109·25-s + 432·29-s − 416·31-s + 412·41-s + 682·49-s − 560·55-s + 740·59-s − 1.10e3·61-s − 1.08e3·71-s + 1.58e3·79-s − 1.87e3·89-s − 192·95-s + 1.18e3·101-s − 740·109-s + 1.20e4·121-s + 936·125-s + 127-s + 131-s + 137-s + 139-s − 1.72e3·145-s + 149-s + 151-s + 1.66e3·155-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 3.83·11-s + 0.579·19-s − 0.871·25-s + 2.76·29-s − 2.41·31-s + 1.56·41-s + 1.98·49-s − 1.37·55-s + 1.63·59-s − 2.30·61-s − 1.80·71-s + 2.25·79-s − 2.23·89-s − 0.207·95-s + 1.16·101-s − 0.650·109-s + 9.04·121-s + 0.669·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.989·145-s + 0.000549·149-s + 0.000538·151-s + 0.862·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.306323592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.306323592\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9342 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14334 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 216 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 208 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 105246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 370 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 550 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 71542 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 540 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 413218 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 792 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 980358 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1822210 T^{2} + p^{6} T^{4} \) |
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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
−10.21494300752468365668362930674, −9.568100281711732189548584610546, −9.316450769193020078576240448388, −9.048788365532477163244870384900, −8.653004967483442220759748084364, −8.200948299399744884715549749707, −7.31706722832136606978202817879, −7.28970608964527741928219389677, −6.73957078026412521026314925313, −6.15067808331699079056410598138, −6.06182631274567230773366555169, −5.34157028449350730733491070418, −4.46593762602424361006896742301, −4.23169931365293033787417860015, −3.77381393481421268474641196108, −3.39744716806499982102542324336, −2.54958695918014148132565777785, −1.66833690159052246119388559745, −1.20951081433644063263851661442, −0.66583347662454195314243970244,
0.66583347662454195314243970244, 1.20951081433644063263851661442, 1.66833690159052246119388559745, 2.54958695918014148132565777785, 3.39744716806499982102542324336, 3.77381393481421268474641196108, 4.23169931365293033787417860015, 4.46593762602424361006896742301, 5.34157028449350730733491070418, 6.06182631274567230773366555169, 6.15067808331699079056410598138, 6.73957078026412521026314925313, 7.28970608964527741928219389677, 7.31706722832136606978202817879, 8.200948299399744884715549749707, 8.653004967483442220759748084364, 9.048788365532477163244870384900, 9.316450769193020078576240448388, 9.568100281711732189548584610546, 10.21494300752468365668362930674
