| L(s) = 1 | − 4·7-s + 6·9-s − 12·17-s − 12·23-s − 25-s + 8·31-s + 4·41-s − 20·47-s − 2·49-s − 24·63-s − 32·71-s + 12·73-s + 27·81-s − 12·89-s + 4·97-s + 36·103-s − 12·113-s + 48·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2·9-s − 2.91·17-s − 2.50·23-s − 1/5·25-s + 1.43·31-s + 0.624·41-s − 2.91·47-s − 2/7·49-s − 3.02·63-s − 3.79·71-s + 1.40·73-s + 3·81-s − 1.27·89-s + 0.406·97-s + 3.54·103-s − 1.12·113-s + 4.40·119-s − 1.27·121-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 − 5.82·153-s + 0.0798·157-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
| \(L(1)\) |
\(\approx\) |
\(0.7904509955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7904509955\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.984901542791837110457225951927, −9.500531220982113530041477837291, −9.238399455313941020356695875689, −8.719445817037772334334958367353, −8.227328185199912112912561955162, −7.81727078378404491332176245625, −7.40531344741522972909192918878, −6.72459723661298358205706366674, −6.64391626858598083899986542118, −6.27467133361255861352315620218, −6.05311844316443491345554064399, −5.06944608532312908599742298206, −4.58950959857587531147014320241, −4.13186499592250019172275169243, −4.11896055521317867858696550904, −3.26370418584880818223236485832, −2.75151687636790418773109998799, −1.88248793834805352888348479246, −1.75794744175765610904034807961, −0.35535869011802554999023140120,
0.35535869011802554999023140120, 1.75794744175765610904034807961, 1.88248793834805352888348479246, 2.75151687636790418773109998799, 3.26370418584880818223236485832, 4.11896055521317867858696550904, 4.13186499592250019172275169243, 4.58950959857587531147014320241, 5.06944608532312908599742298206, 6.05311844316443491345554064399, 6.27467133361255861352315620218, 6.64391626858598083899986542118, 6.72459723661298358205706366674, 7.40531344741522972909192918878, 7.81727078378404491332176245625, 8.227328185199912112912561955162, 8.719445817037772334334958367353, 9.238399455313941020356695875689, 9.500531220982113530041477837291, 9.984901542791837110457225951927
