| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 2·7-s + 8-s + 9-s − 2·12-s + 2·14-s + 16-s − 12·17-s + 18-s − 4·21-s − 2·24-s − 5·25-s + 4·27-s + 2·28-s + 32-s − 12·34-s + 36-s − 4·42-s + 16·43-s − 2·48-s + 3·49-s − 5·50-s + 24·51-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 0.534·14-s + 1/4·16-s − 2.91·17-s + 0.235·18-s − 0.872·21-s − 0.408·24-s − 25-s + 0.769·27-s + 0.377·28-s + 0.176·32-s − 2.05·34-s + 1/6·36-s − 0.617·42-s + 2.43·43-s − 0.288·48-s + 3/7·49-s − 0.707·50-s + 3.36·51-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.533019216248269268759734104285, −7.80365099617056766338064266882, −7.60707532453471338039224955289, −6.71310127772875365703449938682, −6.60348506563741211247253098956, −6.19185902002877876721827038865, −5.38990980084792703269563412867, −5.31921059379060528260176389250, −4.58557293909771964767836479942, −4.19102102405251164216473944262, −3.83664653997120963252560991499, −2.50419401332512614244886579026, −2.39101862347891857041372548490, −1.31920416816215806650160602271, 0,
1.31920416816215806650160602271, 2.39101862347891857041372548490, 2.50419401332512614244886579026, 3.83664653997120963252560991499, 4.19102102405251164216473944262, 4.58557293909771964767836479942, 5.31921059379060528260176389250, 5.38990980084792703269563412867, 6.19185902002877876721827038865, 6.60348506563741211247253098956, 6.71310127772875365703449938682, 7.60707532453471338039224955289, 7.80365099617056766338064266882, 8.533019216248269268759734104285
