| L(s) = 1 | + 3-s − 2·4-s + 5-s + 3·11-s − 2·12-s − 3·13-s + 15-s + 3·16-s + 17-s − 2·20-s − 27-s + 3·33-s − 3·39-s − 6·44-s − 2·47-s + 3·48-s + 51-s + 6·52-s + 3·55-s − 2·60-s − 4·64-s − 3·65-s − 2·68-s − 3·73-s + 2·79-s + 3·80-s − 81-s + ⋯ |
| L(s) = 1 | + 3-s − 2·4-s + 5-s + 3·11-s − 2·12-s − 3·13-s + 15-s + 3·16-s + 17-s − 2·20-s − 27-s + 3·33-s − 3·39-s − 6·44-s − 2·47-s + 3·48-s + 51-s + 6·52-s + 3·55-s − 2·60-s − 4·64-s − 3·65-s − 2·68-s − 3·73-s + 2·79-s + 3·80-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.297266453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297266453\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.572018074871514585228465791321, −9.079145034891776668966499559976, −9.013219015468537170214944870054, −8.380672444201873737322881468988, −8.017752808880501473232286843117, −7.53592962391423997803024212007, −7.35938735459481960594965839927, −6.66702449642579546577166165893, −6.28739442940042629090909078154, −5.83504571601723904767198539355, −5.36015384092720024483420310245, −4.95092895165268069217057530339, −4.62179152519906833082635428373, −4.14850230456053044335796637832, −3.79343513181139495279225106597, −3.12546214183724798141806879621, −3.04414633467805006133123706748, −1.85141798761558520346715506123, −1.83268565553532832818652373202, −0.800219027504191099855463830187,
0.800219027504191099855463830187, 1.83268565553532832818652373202, 1.85141798761558520346715506123, 3.04414633467805006133123706748, 3.12546214183724798141806879621, 3.79343513181139495279225106597, 4.14850230456053044335796637832, 4.62179152519906833082635428373, 4.95092895165268069217057530339, 5.36015384092720024483420310245, 5.83504571601723904767198539355, 6.28739442940042629090909078154, 6.66702449642579546577166165893, 7.35938735459481960594965839927, 7.53592962391423997803024212007, 8.017752808880501473232286843117, 8.380672444201873737322881468988, 9.013219015468537170214944870054, 9.079145034891776668966499559976, 9.572018074871514585228465791321
