| L(s) = 1 | + 4·7-s + 4·13-s − 4·19-s − 4·25-s + 2·37-s + 12·43-s + 2·49-s − 8·61-s + 8·67-s + 6·73-s + 8·79-s + 16·91-s + 14·97-s − 20·103-s + 6·109-s + 8·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.10·13-s − 0.917·19-s − 4/5·25-s + 0.328·37-s + 1.82·43-s + 2/7·49-s − 1.02·61-s + 0.977·67-s + 0.702·73-s + 0.900·79-s + 1.67·91-s + 1.42·97-s − 1.97·103-s + 0.574·109-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.587631992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.587631992\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.117965845767349000385144246676, −7.916862978599173827885363508038, −7.45124827435846047304532781649, −6.87233367513434423874764469745, −6.32084908146833985179775165536, −5.99510368810614965500153829853, −5.49025366834376216054893233753, −4.98721499661367299522041010263, −4.45761727249049658805622833757, −4.08496261187088214791451424382, −3.59571463463586332183810724353, −2.81067293210197952086372659559, −2.07862773707338337245971139521, −1.65519517675676495026349981393, −0.799092026316408552285567257007,
0.799092026316408552285567257007, 1.65519517675676495026349981393, 2.07862773707338337245971139521, 2.81067293210197952086372659559, 3.59571463463586332183810724353, 4.08496261187088214791451424382, 4.45761727249049658805622833757, 4.98721499661367299522041010263, 5.49025366834376216054893233753, 5.99510368810614965500153829853, 6.32084908146833985179775165536, 6.87233367513434423874764469745, 7.45124827435846047304532781649, 7.916862978599173827885363508038, 8.117965845767349000385144246676
