| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 2·13-s + 4·19-s + 4·21-s − 10·25-s + 4·27-s + 4·31-s + 3·37-s − 4·39-s + 10·43-s − 2·49-s − 8·57-s + 4·61-s − 2·63-s + 10·67-s − 20·73-s + 20·75-s − 2·79-s − 11·81-s − 4·91-s − 8·93-s + 16·97-s − 14·103-s + 16·109-s − 6·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.917·19-s + 0.872·21-s − 2·25-s + 0.769·27-s + 0.718·31-s + 0.493·37-s − 0.640·39-s + 1.52·43-s − 2/7·49-s − 1.05·57-s + 0.512·61-s − 0.251·63-s + 1.22·67-s − 2.34·73-s + 2.30·75-s − 0.225·79-s − 1.22·81-s − 0.419·91-s − 0.829·93-s + 1.62·97-s − 1.37·103-s + 1.53·109-s − 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8828399075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8828399075\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.092005347953120115666916676598, −7.67269803382522457997578515915, −7.25382949737628952679589630987, −6.75582503002922924537156608469, −6.19559181337517946617143065027, −6.00201306219100019468368436244, −5.63070976924758382822979381224, −5.14724317669787900181460516532, −4.50663957158533564834122443592, −4.06152599903928441411496520680, −3.49287773767639933470552465701, −2.92484185130011092383794868383, −2.25757760435535767657623074704, −1.31797396963510759984444600515, −0.51217208147971457760390549853,
0.51217208147971457760390549853, 1.31797396963510759984444600515, 2.25757760435535767657623074704, 2.92484185130011092383794868383, 3.49287773767639933470552465701, 4.06152599903928441411496520680, 4.50663957158533564834122443592, 5.14724317669787900181460516532, 5.63070976924758382822979381224, 6.00201306219100019468368436244, 6.19559181337517946617143065027, 6.75582503002922924537156608469, 7.25382949737628952679589630987, 7.67269803382522457997578515915, 8.092005347953120115666916676598
