| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s − 25-s + 4·27-s − 16·31-s + 12·37-s − 8·39-s − 12·41-s − 8·43-s − 6·45-s − 14·49-s − 4·53-s + 8·65-s + 8·67-s − 16·71-s − 2·75-s + 16·79-s + 5·81-s + 8·83-s − 12·89-s − 32·93-s + 24·107-s + 24·111-s − 12·117-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 1/5·25-s + 0.769·27-s − 2.87·31-s + 1.97·37-s − 1.28·39-s − 1.87·41-s − 1.21·43-s − 0.894·45-s − 2·49-s − 0.549·53-s + 0.992·65-s + 0.977·67-s − 1.89·71-s − 0.230·75-s + 1.80·79-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 3.31·93-s + 2.32·107-s + 2.27·111-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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, −8.173282349834764667604075089133, −7.953704755279657121335115603175, −7.26958585488936859387169650645, −7.25136384633542617765663547930, −6.50923348756050362748698725466, −5.87866003990815751467424375923, −5.01357758335939619070346679818, −4.79956865275231776293703222232, −4.02593153952443544832427859826, −3.48889300562368399939117276327, −3.14826068230388029805668446658, −2.23734692694206799493488443258, −1.66440253308003017822578074122, 0,
1.66440253308003017822578074122, 2.23734692694206799493488443258, 3.14826068230388029805668446658, 3.48889300562368399939117276327, 4.02593153952443544832427859826, 4.79956865275231776293703222232, 5.01357758335939619070346679818, 5.87866003990815751467424375923, 6.50923348756050362748698725466, 7.25136384633542617765663547930, 7.26958585488936859387169650645, 7.953704755279657121335115603175, 8.173282349834764667604075089133, 8.681842007730582707537242091206
