| L(s) = 1 | − 3·5-s − 2·7-s + 6·11-s − 5·13-s − 6·17-s + 4·19-s − 6·23-s + 5·25-s − 3·29-s + 4·31-s + 6·35-s + 10·37-s + 6·41-s + 10·43-s + 7·49-s − 12·53-s − 18·55-s + 12·59-s − 5·61-s + 15·65-s − 2·67-s + 12·71-s − 2·73-s − 12·77-s + 10·79-s + 18·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 1.80·11-s − 1.38·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 25-s − 0.557·29-s + 0.718·31-s + 1.01·35-s + 1.64·37-s + 0.937·41-s + 1.52·43-s + 49-s − 1.64·53-s − 2.42·55-s + 1.56·59-s − 0.640·61-s + 1.86·65-s − 0.244·67-s + 1.42·71-s − 0.234·73-s − 1.36·77-s + 1.12·79-s + 1.95·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9120999102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9120999102\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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
−11.66754041152565186880485481359, −11.56229037963041920354744486466, −11.15038007385762031401981620100, −10.49034803084128120866187889568, −9.816204895353720041342496643460, −9.475339924485780284252381223583, −9.155390014874397930741743264054, −8.655620202190949565058895699010, −7.79071996353110318851168208698, −7.67787536090893974604946198624, −7.10471115453247176040541143517, −6.41053879883303465068158051864, −6.30655856326879286692742163827, −5.38715854225702502993223469578, −4.49067354636861185391041141943, −4.20951292402653755898014360815, −3.76327665947738011522374768354, −2.92098375982074329815751560924, −2.17036299705848216660305075705, −0.69069172963674813462339747625,
0.69069172963674813462339747625, 2.17036299705848216660305075705, 2.92098375982074329815751560924, 3.76327665947738011522374768354, 4.20951292402653755898014360815, 4.49067354636861185391041141943, 5.38715854225702502993223469578, 6.30655856326879286692742163827, 6.41053879883303465068158051864, 7.10471115453247176040541143517, 7.67787536090893974604946198624, 7.79071996353110318851168208698, 8.655620202190949565058895699010, 9.155390014874397930741743264054, 9.475339924485780284252381223583, 9.816204895353720041342496643460, 10.49034803084128120866187889568, 11.15038007385762031401981620100, 11.56229037963041920354744486466, 11.66754041152565186880485481359
