| L(s) = 1 | − 4·5-s − 4·7-s + 8·11-s − 3·13-s + 7·17-s + 5·19-s + 8·23-s + 2·25-s − 29-s − 3·31-s + 16·35-s − 11·37-s − 9·41-s + 5·43-s − 3·47-s + 9·49-s + 3·53-s − 32·55-s + 7·59-s − 3·61-s + 12·65-s + 13·67-s − 16·71-s − 7·73-s − 32·77-s − 9·79-s − 83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s + 2.41·11-s − 0.832·13-s + 1.69·17-s + 1.14·19-s + 1.66·23-s + 2/5·25-s − 0.185·29-s − 0.538·31-s + 2.70·35-s − 1.80·37-s − 1.40·41-s + 0.762·43-s − 0.437·47-s + 9/7·49-s + 0.412·53-s − 4.31·55-s + 0.911·59-s − 0.384·61-s + 1.48·65-s + 1.58·67-s − 1.89·71-s − 0.819·73-s − 3.64·77-s − 1.01·79-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9833107845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9833107845\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.181929144747975200575476902291, −8.522349968588605287546419311556, −8.170237791056763088531843273729, −7.63094317241727098709591640360, −7.32602762085257622658210871856, −7.05920867330372511351829035801, −6.86802397872210985318494587672, −6.40581061938038715067589474531, −5.89985238707219584250452345723, −5.40679823602313623260920547324, −5.14236891540570462685648822026, −4.46729936839499916467067204316, −4.06276014963259268542617735246, −3.50432106540420766680754149511, −3.47723461502739637357382598701, −3.27643704619819400499543470219, −2.52665080521329772200878832899, −1.53337384387615373672462624241, −1.12776323559080478535026871174, −0.36559521605923565424899292189,
0.36559521605923565424899292189, 1.12776323559080478535026871174, 1.53337384387615373672462624241, 2.52665080521329772200878832899, 3.27643704619819400499543470219, 3.47723461502739637357382598701, 3.50432106540420766680754149511, 4.06276014963259268542617735246, 4.46729936839499916467067204316, 5.14236891540570462685648822026, 5.40679823602313623260920547324, 5.89985238707219584250452345723, 6.40581061938038715067589474531, 6.86802397872210985318494587672, 7.05920867330372511351829035801, 7.32602762085257622658210871856, 7.63094317241727098709591640360, 8.170237791056763088531843273729, 8.522349968588605287546419311556, 9.181929144747975200575476902291
