L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 4·13-s + 2·17-s + 4·19-s + 21-s − 2·23-s − 27-s + 2·31-s − 33-s − 8·37-s − 4·39-s − 6·41-s − 4·43-s + 49-s − 2·51-s − 6·53-s − 4·57-s + 4·59-s − 6·61-s − 63-s − 4·67-s + 2·69-s + 4·73-s − 77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s + 0.359·31-s − 0.174·33-s − 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.240·69-s + 0.468·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.737068666\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.737068666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38574977092651, −11.98085375567547, −11.68382657586658, −11.21413331581183, −10.63151265200419, −10.30313639263221, −9.898380884841526, −9.285586828786696, −8.987001792340950, −8.359039239444276, −7.919175433395191, −7.450514851914359, −6.795532406418576, −6.492789666152273, −6.043978615380578, −5.521022277236809, −5.053026218807198, −4.610629404673798, −3.777078937797023, −3.532222400021667, −3.084226726367865, −2.216163501813192, −1.553549185610789, −1.118550265683077, −0.3850848401828601,
0.3850848401828601, 1.118550265683077, 1.553549185610789, 2.216163501813192, 3.084226726367865, 3.532222400021667, 3.777078937797023, 4.610629404673798, 5.053026218807198, 5.521022277236809, 6.043978615380578, 6.492789666152273, 6.795532406418576, 7.450514851914359, 7.919175433395191, 8.359039239444276, 8.987001792340950, 9.285586828786696, 9.898380884841526, 10.30313639263221, 10.63151265200419, 11.21413331581183, 11.68382657586658, 11.98085375567547, 12.38574977092651