| L(s) = 1 | + 2·3-s − 8·5-s + 3·9-s − 3·11-s − 16·15-s − 4·16-s − 12·23-s + 38·25-s + 4·27-s + 14·31-s − 6·33-s − 14·37-s − 24·45-s + 6·47-s − 8·48-s − 10·49-s − 12·53-s + 24·55-s − 4·67-s − 24·69-s − 6·71-s + 76·75-s + 32·80-s + 5·81-s − 20·89-s + 28·93-s − 24·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3.57·5-s + 9-s − 0.904·11-s − 4.13·15-s − 16-s − 2.50·23-s + 38/5·25-s + 0.769·27-s + 2.51·31-s − 1.04·33-s − 2.30·37-s − 3.57·45-s + 0.875·47-s − 1.15·48-s − 1.42·49-s − 1.64·53-s + 3.23·55-s − 0.488·67-s − 2.88·69-s − 0.712·71-s + 8.77·75-s + 3.57·80-s + 5/9·81-s − 2.11·89-s + 2.90·93-s − 2.43·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1830609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1830609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−7.75369551411276366117982677734, −7.04855950160821631603912974995, −6.86675936068597319462443271492, −6.38233044704197809374592113660, −5.52011843602638672484358381274, −4.72107582112055310584779205405, −4.53745508909086524538385836361, −4.20785129175750916654234507255, −3.75520298510552500673702713548, −3.31716866309577067376325912744, −2.91545403229970325971647429959, −2.37543498218440605661981594235, −1.36201747210325541983004487330, 0, 0,
1.36201747210325541983004487330, 2.37543498218440605661981594235, 2.91545403229970325971647429959, 3.31716866309577067376325912744, 3.75520298510552500673702713548, 4.20785129175750916654234507255, 4.53745508909086524538385836361, 4.72107582112055310584779205405, 5.52011843602638672484358381274, 6.38233044704197809374592113660, 6.86675936068597319462443271492, 7.04855950160821631603912974995, 7.75369551411276366117982677734
