| L(s) = 1 | − 2·3-s − 4·7-s + 3·9-s + 11-s + 6·13-s − 2·17-s + 5·19-s + 8·21-s + 7·23-s − 10·27-s − 12·29-s − 4·31-s − 2·33-s − 5·37-s − 12·39-s − 10·41-s − 12·43-s − 9·47-s + 9·49-s + 4·51-s + 11·53-s − 10·57-s − 8·59-s + 12·61-s − 12·63-s − 4·67-s − 14·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 9-s + 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.14·19-s + 1.74·21-s + 1.45·23-s − 1.92·27-s − 2.22·29-s − 0.718·31-s − 0.348·33-s − 0.821·37-s − 1.92·39-s − 1.56·41-s − 1.82·43-s − 1.31·47-s + 9/7·49-s + 0.560·51-s + 1.51·53-s − 1.32·57-s − 1.04·59-s + 1.53·61-s − 1.51·63-s − 0.488·67-s − 1.68·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5429231206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5429231206\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08921634039807693777943196528, −9.227195249067869121867676290485, −9.218232201703958353331722765946, −8.686558791093811603843156092851, −8.213883586280911004510475687181, −7.53348643410263489050637756703, −7.16885078576524821138669290795, −6.80617617349166798761665443841, −6.51431378284343142968756851234, −6.11255172395839609847995573404, −5.59701052952267362423131515776, −5.24381242950577318557947562524, −5.04317126478487142612273099994, −3.94112991422782330077792513114, −3.78701812917989871350363925335, −3.42593979867028206654495862237, −2.87048462374737838085946338544, −1.72741785130060712862589600306, −1.43443711147013029053496661736, −0.33464136762397425420927785505,
0.33464136762397425420927785505, 1.43443711147013029053496661736, 1.72741785130060712862589600306, 2.87048462374737838085946338544, 3.42593979867028206654495862237, 3.78701812917989871350363925335, 3.94112991422782330077792513114, 5.04317126478487142612273099994, 5.24381242950577318557947562524, 5.59701052952267362423131515776, 6.11255172395839609847995573404, 6.51431378284343142968756851234, 6.80617617349166798761665443841, 7.16885078576524821138669290795, 7.53348643410263489050637756703, 8.213883586280911004510475687181, 8.686558791093811603843156092851, 9.218232201703958353331722765946, 9.227195249067869121867676290485, 10.08921634039807693777943196528
