| L(s) = 1 | − 4·5-s − 6·9-s + 8·13-s + 11·25-s + 24·37-s + 20·41-s + 24·45-s + 14·49-s + 8·53-s − 32·65-s + 27·81-s − 20·89-s − 48·117-s + 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2·9-s + 2.21·13-s + 11/5·25-s + 3.94·37-s + 3.12·41-s + 3.57·45-s + 2·49-s + 1.09·53-s − 3.96·65-s + 3·81-s − 2.11·89-s − 4.43·117-s + 2·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.537338284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537338284\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−9.615574314923288353865570939568, −9.429711402717261654722650621073, −8.835687006138474121852220006503, −8.577254095764506108823686427941, −8.308118398786823912895194295791, −7.961190862648714773199738838735, −7.48212053139456271821073999341, −7.25242257584491243425823245549, −6.37581671780996700173250747470, −6.09564992246313989759138835603, −5.79976791203005980635678570278, −5.43442945106864257347401247849, −4.43026559039929634531679062070, −4.27959776292931403413698728579, −3.83337270907240995283737323352, −3.34422504374411164713378759684, −2.66161351336519612778498726509, −2.54437320042756666156764161848, −0.953118674934607983376915718507, −0.71385340019705987594688371827,
0.71385340019705987594688371827, 0.953118674934607983376915718507, 2.54437320042756666156764161848, 2.66161351336519612778498726509, 3.34422504374411164713378759684, 3.83337270907240995283737323352, 4.27959776292931403413698728579, 4.43026559039929634531679062070, 5.43442945106864257347401247849, 5.79976791203005980635678570278, 6.09564992246313989759138835603, 6.37581671780996700173250747470, 7.25242257584491243425823245549, 7.48212053139456271821073999341, 7.961190862648714773199738838735, 8.308118398786823912895194295791, 8.577254095764506108823686427941, 8.835687006138474121852220006503, 9.429711402717261654722650621073, 9.615574314923288353865570939568
