L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s + 2·11-s + 4·14-s + 5·16-s + 4·22-s − 12·23-s − 10·25-s + 6·28-s − 12·29-s + 6·32-s − 20·37-s + 16·43-s + 6·44-s − 24·46-s − 3·49-s − 20·50-s + 8·56-s − 24·58-s + 7·64-s − 8·67-s − 12·71-s − 40·74-s + 4·77-s + 28·79-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s + 0.603·11-s + 1.06·14-s + 5/4·16-s + 0.852·22-s − 2.50·23-s − 2·25-s + 1.13·28-s − 2.22·29-s + 1.06·32-s − 3.28·37-s + 2.43·43-s + 0.904·44-s − 3.53·46-s − 3/7·49-s − 2.82·50-s + 1.06·56-s − 3.15·58-s + 7/8·64-s − 0.977·67-s − 1.42·71-s − 4.64·74-s + 0.455·77-s + 3.15·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.37562026200620347083900898682, −7.29087986975847329500837910780, −6.52504829438381665573881626069, −6.01365350861836338965349792477, −5.85955462295020247251479297795, −5.44687910527563212016596341412, −4.96524457370890568674874182561, −4.36376964393171453387912321319, −3.89761457992935995656155838511, −3.77672609499705967583268412266, −3.25261158060160886155862976042, −2.12020721928430735862314078108, −2.06167323195037554066088312898, −1.54124369917794574272131224237, 0,
1.54124369917794574272131224237, 2.06167323195037554066088312898, 2.12020721928430735862314078108, 3.25261158060160886155862976042, 3.77672609499705967583268412266, 3.89761457992935995656155838511, 4.36376964393171453387912321319, 4.96524457370890568674874182561, 5.44687910527563212016596341412, 5.85955462295020247251479297795, 6.01365350861836338965349792477, 6.52504829438381665573881626069, 7.29087986975847329500837910780, 7.37562026200620347083900898682