| L(s) = 1 | + 3-s + 3·7-s + 9-s − 4·16-s − 4·17-s + 3·21-s + 27-s − 4·37-s − 16·41-s − 2·43-s − 4·47-s − 4·48-s + 2·49-s − 4·51-s − 20·59-s + 3·63-s + 6·67-s + 81-s − 12·83-s + 24·101-s + 10·109-s − 4·111-s − 12·112-s − 12·119-s − 18·121-s − 16·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s − 16-s − 0.970·17-s + 0.654·21-s + 0.192·27-s − 0.657·37-s − 2.49·41-s − 0.304·43-s − 0.583·47-s − 0.577·48-s + 2/7·49-s − 0.560·51-s − 2.60·59-s + 0.377·63-s + 0.733·67-s + 1/9·81-s − 1.31·83-s + 2.38·101-s + 0.957·109-s − 0.379·111-s − 1.13·112-s − 1.10·119-s − 1.63·121-s − 1.44·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.114757177440921419069207299350, −7.57708995620639843061937721934, −7.16093151076970733727873825913, −6.77287251257420576381216593643, −6.31467516859974173325654268001, −5.77288972312895551005089105172, −4.97413968254612633796613688199, −4.76969969138667681909645581059, −4.48018036005395779311676580836, −3.64025397588061515675536757443, −3.26639824967443527361709363812, −2.46120401788391367367701831388, −1.90319955654708104595670502735, −1.46977720012697694390681972111, 0,
1.46977720012697694390681972111, 1.90319955654708104595670502735, 2.46120401788391367367701831388, 3.26639824967443527361709363812, 3.64025397588061515675536757443, 4.48018036005395779311676580836, 4.76969969138667681909645581059, 4.97413968254612633796613688199, 5.77288972312895551005089105172, 6.31467516859974173325654268001, 6.77287251257420576381216593643, 7.16093151076970733727873825913, 7.57708995620639843061937721934, 8.114757177440921419069207299350
