| L(s) = 1 | + 4·5-s − 2·7-s − 2·13-s − 12·23-s + 2·25-s + 8·29-s − 8·35-s − 11·49-s − 4·53-s − 16·59-s − 8·65-s − 10·67-s − 28·71-s − 8·83-s + 4·91-s + 16·103-s + 20·107-s + 18·109-s − 48·115-s + 9·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 0.554·13-s − 2.50·23-s + 2/5·25-s + 1.48·29-s − 1.35·35-s − 1.57·49-s − 0.549·53-s − 2.08·59-s − 0.992·65-s − 1.22·67-s − 3.32·71-s − 0.878·83-s + 0.419·91-s + 1.57·103-s + 1.93·107-s + 1.72·109-s − 4.47·115-s + 9/11·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17438976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17438976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9771253818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9771253818\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 29 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
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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
−8.801193033881346934726583575770, −8.315834039062301716376174880823, −7.76193278202135829803815258937, −7.38045420883397372181333078335, −7.34746150520247091680711571109, −6.42579009821763858050626637711, −6.18598865285740576877548052320, −6.06843794794772977878804490039, −6.02623780266420029379472670977, −5.26305002210859246140982030715, −4.93418705863171696024514206856, −4.37680998764650945137570617182, −4.25544655395146962191472345705, −3.37054819610790891935477746635, −3.19431220740153337371971015114, −2.58227060630973549358423372858, −2.23034490107505268348732896592, −1.61658495578808948874604814716, −1.49829512237434322372300153158, −0.25513166186267830678442102945,
0.25513166186267830678442102945, 1.49829512237434322372300153158, 1.61658495578808948874604814716, 2.23034490107505268348732896592, 2.58227060630973549358423372858, 3.19431220740153337371971015114, 3.37054819610790891935477746635, 4.25544655395146962191472345705, 4.37680998764650945137570617182, 4.93418705863171696024514206856, 5.26305002210859246140982030715, 6.02623780266420029379472670977, 6.06843794794772977878804490039, 6.18598865285740576877548052320, 6.42579009821763858050626637711, 7.34746150520247091680711571109, 7.38045420883397372181333078335, 7.76193278202135829803815258937, 8.315834039062301716376174880823, 8.801193033881346934726583575770
