| L(s) = 1 | + 2·3-s + 9-s + 4·11-s − 12·13-s + 8·23-s − 10·25-s − 4·27-s + 8·33-s − 8·37-s − 24·39-s − 16·47-s − 14·49-s + 24·61-s + 16·69-s + 24·71-s + 4·73-s − 20·75-s − 11·81-s + 32·83-s − 36·97-s + 4·99-s − 36·107-s − 16·111-s − 12·117-s − 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s − 3.32·13-s + 1.66·23-s − 2·25-s − 0.769·27-s + 1.39·33-s − 1.31·37-s − 3.84·39-s − 2.33·47-s − 2·49-s + 3.07·61-s + 1.92·69-s + 2.84·71-s + 0.468·73-s − 2.30·75-s − 1.22·81-s + 3.51·83-s − 3.65·97-s + 0.402·99-s − 3.48·107-s − 1.51·111-s − 1.10·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−8.422820303858929928484512856269, −7.918822763878182275157603554744, −7.915799644978553160956414624199, −7.04028809245781208083955490002, −6.84063454436614469172097574333, −6.48858014013781204324172880722, −5.27740412064621881845926355386, −5.22026991260055872535309051671, −4.70132376102771618663878285353, −3.69893087635466446387661413772, −3.65040990576825303475268770324, −2.65731672220088998433790242398, −2.33145296998291404024563148829, −1.62565942265304447284805954301, 0,
1.62565942265304447284805954301, 2.33145296998291404024563148829, 2.65731672220088998433790242398, 3.65040990576825303475268770324, 3.69893087635466446387661413772, 4.70132376102771618663878285353, 5.22026991260055872535309051671, 5.27740412064621881845926355386, 6.48858014013781204324172880722, 6.84063454436614469172097574333, 7.04028809245781208083955490002, 7.915799644978553160956414624199, 7.918822763878182275157603554744, 8.422820303858929928484512856269
