| L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s − 4·16-s + 12·23-s − 4·27-s − 6·31-s − 4·33-s + 4·37-s + 4·47-s + 8·48-s − 5·49-s − 8·53-s − 20·59-s − 6·67-s − 24·69-s − 16·71-s + 5·81-s + 12·93-s + 34·97-s + 6·99-s − 8·103-s − 8·111-s − 8·113-s − 7·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.603·11-s − 16-s + 2.50·23-s − 0.769·27-s − 1.07·31-s − 0.696·33-s + 0.657·37-s + 0.583·47-s + 1.15·48-s − 5/7·49-s − 1.09·53-s − 2.60·59-s − 0.733·67-s − 2.88·69-s − 1.89·71-s + 5/9·81-s + 1.24·93-s + 3.45·97-s + 0.603·99-s − 0.788·103-s − 0.759·111-s − 0.752·113-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−7.84902432953827439364529578981, −7.63445690246110755414862928105, −7.06328550761489343142245284993, −6.79375443883521821659151920822, −6.16165631359848537381830780833, −6.05569215012514529764032801313, −5.26550699194046601552455316284, −4.80645524398487230267862763673, −4.62309319523854591570832526017, −3.93276270400122129280671872839, −3.24466149791278329969031792942, −2.70866001385341573464253183059, −1.71419845103506630043143205825, −1.13781757996508329611151326735, 0,
1.13781757996508329611151326735, 1.71419845103506630043143205825, 2.70866001385341573464253183059, 3.24466149791278329969031792942, 3.93276270400122129280671872839, 4.62309319523854591570832526017, 4.80645524398487230267862763673, 5.26550699194046601552455316284, 6.05569215012514529764032801313, 6.16165631359848537381830780833, 6.79375443883521821659151920822, 7.06328550761489343142245284993, 7.63445690246110755414862928105, 7.84902432953827439364529578981
