| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s − 5·9-s − 10·11-s − 2·14-s + 5·16-s + 10·18-s + 20·22-s + 8·23-s − 6·25-s + 3·28-s − 6·29-s − 6·32-s − 15·36-s + 2·37-s + 16·43-s − 30·44-s − 16·46-s − 6·49-s + 12·50-s − 28·53-s − 4·56-s + 12·58-s − 5·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 5/3·9-s − 3.01·11-s − 0.534·14-s + 5/4·16-s + 2.35·18-s + 4.26·22-s + 1.66·23-s − 6/5·25-s + 0.566·28-s − 1.11·29-s − 1.06·32-s − 5/2·36-s + 0.328·37-s + 2.43·43-s − 4.52·44-s − 2.35·46-s − 6/7·49-s + 1.69·50-s − 3.84·53-s − 0.534·56-s + 1.57·58-s − 0.629·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−7.73194439801498984990067036036, −7.36625732552646488198525538003, −6.95941542576258623411341914340, −5.98145963357035298109695620640, −5.88341646402151712569560354820, −5.58762621523697466604199552504, −4.88572076999254851436349526862, −4.65218853180790869047123049112, −3.48227040412980528111791240591, −2.94234828226630578092715290383, −2.69630329480699135407030849925, −2.21613701544165268554352237565, −1.35839013518898789937877263854, 0, 0,
1.35839013518898789937877263854, 2.21613701544165268554352237565, 2.69630329480699135407030849925, 2.94234828226630578092715290383, 3.48227040412980528111791240591, 4.65218853180790869047123049112, 4.88572076999254851436349526862, 5.58762621523697466604199552504, 5.88341646402151712569560354820, 5.98145963357035298109695620640, 6.95941542576258623411341914340, 7.36625732552646488198525538003, 7.73194439801498984990067036036
