L(s) = 1 | − 3-s + 2·5-s + 9-s + 11-s − 13-s − 2·15-s − 6·17-s + 4·19-s − 8·23-s − 25-s − 27-s + 10·29-s − 33-s − 6·37-s + 39-s + 10·41-s − 4·43-s + 2·45-s + 8·47-s − 7·49-s + 6·51-s + 10·53-s + 2·55-s − 4·57-s + 12·59-s − 14·61-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.174·33-s − 0.986·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.840·51-s + 1.37·53-s + 0.269·55-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.827260165\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.827260165\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.49800447133753, −14.50783117053532, −14.17908335654713, −13.51619575883578, −13.36952096265459, −12.40628560502106, −12.04900775672564, −11.60978857171258, −10.81219427056809, −10.43831787413497, −9.729474150804866, −9.512652991724287, −8.678043459357197, −8.176977636476881, −7.361781113889509, −6.725764869751797, −6.322531364791525, −5.669110781037255, −5.206106599215582, −4.361414791045512, −3.971023798320213, −2.836625904733328, −2.218630383868047, −1.521837176153214, −0.5456529863658735,
0.5456529863658735, 1.521837176153214, 2.218630383868047, 2.836625904733328, 3.971023798320213, 4.361414791045512, 5.206106599215582, 5.669110781037255, 6.322531364791525, 6.725764869751797, 7.361781113889509, 8.176977636476881, 8.678043459357197, 9.512652991724287, 9.729474150804866, 10.43831787413497, 10.81219427056809, 11.60978857171258, 12.04900775672564, 12.40628560502106, 13.36952096265459, 13.51619575883578, 14.17908335654713, 14.50783117053532, 15.49800447133753