| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 9-s − 6·12-s + 5·16-s − 12·17-s + 2·18-s − 8·24-s − 10·25-s + 4·27-s + 12·29-s − 8·31-s + 6·32-s − 24·34-s + 3·36-s + 4·37-s − 12·41-s − 10·48-s + 49-s − 20·50-s + 24·51-s + 8·54-s + 24·58-s − 16·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 1/3·9-s − 1.73·12-s + 5/4·16-s − 2.91·17-s + 0.471·18-s − 1.63·24-s − 2·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s + 1.06·32-s − 4.11·34-s + 1/2·36-s + 0.657·37-s − 1.87·41-s − 1.44·48-s + 1/7·49-s − 2.82·50-s + 3.36·51-s + 1.08·54-s + 3.15·58-s − 2.03·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.888226006934558931566258496787, −8.145906800879012290031802042193, −7.80365099617056766338064266882, −6.82397964191468499058519060303, −6.71310127772875365703449938682, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −5.31921059379060528260176389250, −4.85990751466979898816213085515, −4.19102102405251164216473944262, −4.09052365620160024762211243831, −3.03039761753313177141617611476, −2.39101862347891857041372548490, −1.67120553980965395706477292535, 0,
1.67120553980965395706477292535, 2.39101862347891857041372548490, 3.03039761753313177141617611476, 4.09052365620160024762211243831, 4.19102102405251164216473944262, 4.85990751466979898816213085515, 5.31921059379060528260176389250, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.71310127772875365703449938682, 6.82397964191468499058519060303, 7.80365099617056766338064266882, 8.145906800879012290031802042193, 8.888226006934558931566258496787
