| L(s) = 1 | + 3-s + 2·5-s + 9-s − 4·13-s + 2·15-s − 4·17-s − 8·19-s − 25-s + 27-s − 12·29-s + 12·37-s − 4·39-s + 2·45-s − 14·49-s − 4·51-s − 8·57-s − 8·65-s − 16·71-s − 75-s + 81-s + 8·83-s − 8·85-s − 12·87-s − 16·95-s + 36·101-s + 32·103-s + 24·107-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.10·13-s + 0.516·15-s − 0.970·17-s − 1.83·19-s − 1/5·25-s + 0.192·27-s − 2.22·29-s + 1.97·37-s − 0.640·39-s + 0.298·45-s − 2·49-s − 0.560·51-s − 1.05·57-s − 0.992·65-s − 1.89·71-s − 0.115·75-s + 1/9·81-s + 0.878·83-s − 0.867·85-s − 1.28·87-s − 1.64·95-s + 3.58·101-s + 3.15·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.681842007730582707537242091206, −7.953704755279657121335115603175, −7.62403207119791885504595168937, −7.25136384633542617765663547930, −6.50923348756050362748698725466, −6.14969218484336320739052245813, −5.87866003990815751467424375923, −4.83326855247570334999188527992, −4.79956865275231776293703222232, −4.02593153952443544832427859826, −3.48889300562368399939117276327, −2.51659494204614894722398853037, −2.23734692694206799493488443258, −1.66440253308003017822578074122, 0,
1.66440253308003017822578074122, 2.23734692694206799493488443258, 2.51659494204614894722398853037, 3.48889300562368399939117276327, 4.02593153952443544832427859826, 4.79956865275231776293703222232, 4.83326855247570334999188527992, 5.87866003990815751467424375923, 6.14969218484336320739052245813, 6.50923348756050362748698725466, 7.25136384633542617765663547930, 7.62403207119791885504595168937, 7.953704755279657121335115603175, 8.681842007730582707537242091206
