| L(s) = 1 | − 2·4-s + 12·13-s + 4·16-s − 8·25-s − 4·37-s − 24·52-s + 24·61-s − 8·64-s − 12·73-s − 36·97-s + 16·100-s − 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s + 3.32·13-s + 16-s − 8/5·25-s − 0.657·37-s − 3.32·52-s + 3.07·61-s − 64-s − 1.40·73-s − 3.65·97-s + 8/5·100-s − 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.842474238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842474238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−9.652173553832694733360080910155, −8.873119540056533245525347234187, −8.801394743501445615550369381552, −8.358622308332705733988312252279, −7.937538777983804333700619065598, −7.916842880453020762746270948022, −6.98576482890637229015149049786, −6.66046356916649474593699749730, −6.26884477414532737926062521435, −5.77045337030886139303586056636, −5.37875314578908481358825924015, −5.31136998390671865165742300088, −4.16319772384089492977424346533, −4.03822964536088680837124253099, −3.89160305151847577774218133684, −3.24958931687781157788739063697, −2.71113413995048840570646202076, −1.64625240467082765857055783044, −1.40433213360464605806209832799, −0.56543345258875710026540621223,
0.56543345258875710026540621223, 1.40433213360464605806209832799, 1.64625240467082765857055783044, 2.71113413995048840570646202076, 3.24958931687781157788739063697, 3.89160305151847577774218133684, 4.03822964536088680837124253099, 4.16319772384089492977424346533, 5.31136998390671865165742300088, 5.37875314578908481358825924015, 5.77045337030886139303586056636, 6.26884477414532737926062521435, 6.66046356916649474593699749730, 6.98576482890637229015149049786, 7.916842880453020762746270948022, 7.937538777983804333700619065598, 8.358622308332705733988312252279, 8.801394743501445615550369381552, 8.873119540056533245525347234187, 9.652173553832694733360080910155
