| L(s) = 1 | − 2·3-s + 9-s − 6·11-s − 8·13-s − 9·25-s + 4·27-s + 12·33-s − 20·37-s + 16·39-s − 18·47-s − 5·49-s + 28·59-s − 10·61-s − 12·71-s − 30·73-s + 18·75-s − 11·81-s + 8·83-s + 32·97-s − 6·99-s + 20·107-s + 24·109-s + 40·111-s − 8·117-s + 5·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.80·11-s − 2.21·13-s − 9/5·25-s + 0.769·27-s + 2.08·33-s − 3.28·37-s + 2.56·39-s − 2.62·47-s − 5/7·49-s + 3.64·59-s − 1.28·61-s − 1.42·71-s − 3.51·73-s + 2.07·75-s − 1.22·81-s + 0.878·83-s + 3.24·97-s − 0.603·99-s + 1.93·107-s + 2.29·109-s + 3.79·111-s − 0.739·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−8.148422841499315186159351254036, −7.53818989363645791606962986645, −7.20024501520191211813153160377, −6.97775492717497319205239152750, −6.01501469952133530659188791248, −5.89152757780490661000300260672, −5.15976497091815013069726027465, −4.85172758929064856672456822692, −4.78246098563165761782711476378, −3.62617557238284773485534023353, −3.08915058635532111035563541614, −2.33036552779137826129420876510, −1.81707434889353167746748104685, 0, 0,
1.81707434889353167746748104685, 2.33036552779137826129420876510, 3.08915058635532111035563541614, 3.62617557238284773485534023353, 4.78246098563165761782711476378, 4.85172758929064856672456822692, 5.15976497091815013069726027465, 5.89152757780490661000300260672, 6.01501469952133530659188791248, 6.97775492717497319205239152750, 7.20024501520191211813153160377, 7.53818989363645791606962986645, 8.148422841499315186159351254036
