| L(s) = 1 | − 2·5-s − 9-s + 16·19-s − 25-s − 12·29-s + 16·31-s + 12·41-s + 2·45-s − 2·49-s + 12·61-s + 32·71-s − 16·79-s + 81-s + 20·89-s − 32·95-s − 28·101-s − 20·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s + 3.67·19-s − 1/5·25-s − 2.22·29-s + 2.87·31-s + 1.87·41-s + 0.298·45-s − 2/7·49-s + 1.53·61-s + 3.79·71-s − 1.80·79-s + 1/9·81-s + 2.11·89-s − 3.28·95-s − 2.78·101-s − 1.91·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818502763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818502763\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.15171153648370455755473970473, −9.650792749302337946181799221801, −9.401050543851123561989668694038, −9.262222206835457484755614124135, −8.415103588954964488254877030404, −7.968597947754405242181241195371, −7.71240976081536572422566947413, −7.58190292405612893024544673975, −6.70866737316784786177296608315, −6.68985947927623828947056456137, −5.60530398770157714774534466327, −5.59818820227675093164261046348, −5.09376829222111799629919852001, −4.46234821772170555779770077073, −3.75854924307519632440789484748, −3.62190216826083507701274624659, −2.85035100503354657465662311055, −2.48135447738237884783741304560, −1.29914658144467961439861519154, −0.71825506655189134218654800495,
0.71825506655189134218654800495, 1.29914658144467961439861519154, 2.48135447738237884783741304560, 2.85035100503354657465662311055, 3.62190216826083507701274624659, 3.75854924307519632440789484748, 4.46234821772170555779770077073, 5.09376829222111799629919852001, 5.59818820227675093164261046348, 5.60530398770157714774534466327, 6.68985947927623828947056456137, 6.70866737316784786177296608315, 7.58190292405612893024544673975, 7.71240976081536572422566947413, 7.968597947754405242181241195371, 8.415103588954964488254877030404, 9.262222206835457484755614124135, 9.401050543851123561989668694038, 9.650792749302337946181799221801, 10.15171153648370455755473970473
