| L(s) = 1 | − 7-s + 6·11-s − 2·13-s − 8·19-s + 6·23-s + 5·25-s − 6·29-s − 8·31-s + 4·37-s − 12·41-s + 4·43-s − 12·47-s − 12·53-s + 10·61-s − 8·67-s + 12·71-s − 20·73-s − 6·77-s + 4·79-s + 12·83-s + 24·89-s + 2·91-s + 10·97-s + 12·101-s − 8·103-s − 12·107-s + 28·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.554·13-s − 1.83·19-s + 1.25·23-s + 25-s − 1.11·29-s − 1.43·31-s + 0.657·37-s − 1.87·41-s + 0.609·43-s − 1.75·47-s − 1.64·53-s + 1.28·61-s − 0.977·67-s + 1.42·71-s − 2.34·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s + 2.54·89-s + 0.209·91-s + 1.01·97-s + 1.19·101-s − 0.788·103-s − 1.16·107-s + 2.68·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.599163012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.599163012\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.277514125400977374924380205898, −8.916673708942122618435374206025, −8.581633392470210362959985492589, −8.139114171419124658855805662152, −7.55487425682306110698043040046, −7.26718065801961995198100705756, −6.80280541089586687415480976047, −6.45421799154760728205789787926, −6.31189553496052650422038856896, −5.80930879213329643422588999269, −5.04670819917669227594888150587, −4.87651503975233622159056832727, −4.46884447186603265294106039120, −3.71241073276804783745447111211, −3.61902159725980669399051113580, −3.11662560351249626213377973979, −2.36346957374264415656614906676, −1.83537165811994897304706491537, −1.39522684320367510089473238635, −0.43932082541865337390556094970,
0.43932082541865337390556094970, 1.39522684320367510089473238635, 1.83537165811994897304706491537, 2.36346957374264415656614906676, 3.11662560351249626213377973979, 3.61902159725980669399051113580, 3.71241073276804783745447111211, 4.46884447186603265294106039120, 4.87651503975233622159056832727, 5.04670819917669227594888150587, 5.80930879213329643422588999269, 6.31189553496052650422038856896, 6.45421799154760728205789787926, 6.80280541089586687415480976047, 7.26718065801961995198100705756, 7.55487425682306110698043040046, 8.139114171419124658855805662152, 8.581633392470210362959985492589, 8.916673708942122618435374206025, 9.277514125400977374924380205898
