| L(s) = 1 | − 7-s − 3·9-s − 3·11-s − 13-s − 12·17-s − 8·19-s − 3·23-s − 3·29-s − 5·31-s − 4·37-s − 3·41-s − 43-s − 9·47-s + 7·49-s + 12·53-s + 3·59-s + 13·61-s + 3·63-s − 7·67-s − 24·71-s + 20·73-s + 3·77-s − 11·79-s + 9·81-s − 9·83-s + 12·89-s + 91-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.904·11-s − 0.277·13-s − 2.91·17-s − 1.83·19-s − 0.625·23-s − 0.557·29-s − 0.898·31-s − 0.657·37-s − 0.468·41-s − 0.152·43-s − 1.31·47-s + 49-s + 1.64·53-s + 0.390·59-s + 1.66·61-s + 0.377·63-s − 0.855·67-s − 2.84·71-s + 2.34·73-s + 0.341·77-s − 1.23·79-s + 81-s − 0.987·83-s + 1.27·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.934027865672006188887462067736, −9.446450082304142950428623007047, −8.824464568938342661037299540489, −8.657535676838555908508705269218, −8.495590542006043369278903731983, −7.87755598588874946074487982599, −7.17898116192548275513020595166, −6.96871749481446943556934251209, −6.32588555767122081102852786984, −6.21201664046168837127418418978, −5.31540893431529251084337468439, −5.29131787625448468559218736954, −4.32098684488277843466422901071, −4.21299150744187058111330894299, −3.51113437402413471268308823176, −2.68360377740080614410689010106, −2.29134600621848013287689897664, −1.92110189704250879355049240467, 0, 0,
1.92110189704250879355049240467, 2.29134600621848013287689897664, 2.68360377740080614410689010106, 3.51113437402413471268308823176, 4.21299150744187058111330894299, 4.32098684488277843466422901071, 5.29131787625448468559218736954, 5.31540893431529251084337468439, 6.21201664046168837127418418978, 6.32588555767122081102852786984, 6.96871749481446943556934251209, 7.17898116192548275513020595166, 7.87755598588874946074487982599, 8.495590542006043369278903731983, 8.657535676838555908508705269218, 8.824464568938342661037299540489, 9.446450082304142950428623007047, 9.934027865672006188887462067736
