| L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 4·13-s + 8·23-s − 6·25-s − 4·27-s − 8·33-s + 20·37-s + 8·39-s − 16·47-s + 2·49-s + 28·59-s + 4·61-s + 16·69-s + 24·71-s + 28·73-s − 12·75-s − 11·81-s − 12·83-s − 4·97-s − 4·99-s − 4·107-s − 12·109-s + 40·111-s + 4·117-s − 10·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 1.66·23-s − 6/5·25-s − 0.769·27-s − 1.39·33-s + 3.28·37-s + 1.28·39-s − 2.33·47-s + 2/7·49-s + 3.64·59-s + 0.512·61-s + 1.92·69-s + 2.84·71-s + 3.27·73-s − 1.38·75-s − 1.22·81-s − 1.31·83-s − 0.406·97-s − 0.402·99-s − 0.386·107-s − 1.14·109-s + 3.79·111-s + 0.369·117-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351695258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351695258\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.385355573475711088007535446099, −8.675368536957604275874860747441, −8.173183686627627779120693205031, −8.088659191412945349881968287052, −7.63242500478736670765641851926, −6.78571919296342801026858814862, −6.52256885432452015704757780960, −5.64130249043000575312616744005, −5.34050418818343429420051907291, −4.62995407509459210959268189869, −3.73824894896581631226090928700, −3.56276783709589056726776122484, −2.46608091328452001690059553408, −2.44235807183272353930685105042, −1.02724387350145696940295704376,
1.02724387350145696940295704376, 2.44235807183272353930685105042, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 3.73824894896581631226090928700, 4.62995407509459210959268189869, 5.34050418818343429420051907291, 5.64130249043000575312616744005, 6.52256885432452015704757780960, 6.78571919296342801026858814862, 7.63242500478736670765641851926, 8.088659191412945349881968287052, 8.173183686627627779120693205031, 8.675368536957604275874860747441, 9.385355573475711088007535446099
