L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 13-s − 15-s + 17-s + 4·19-s + 25-s − 27-s − 2·29-s + 4·33-s + 6·37-s − 39-s − 6·41-s + 4·43-s + 45-s − 7·49-s − 51-s + 6·53-s − 4·55-s − 4·57-s − 4·59-s + 6·61-s + 65-s − 12·67-s + 16·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 49-s − 0.140·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.124·65-s − 1.46·67-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.683038219\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683038219\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
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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
−14.26253144566799, −14.00955253649703, −13.26584602953944, −12.97736127477390, −12.53806558087557, −11.78205123845034, −11.38488051570087, −10.91385561862521, −10.18158354496041, −10.02550692501579, −9.388214716413399, −8.705785076276377, −8.166960425121719, −7.412544286663916, −7.263769946195125, −6.248794451030042, −5.971844822688002, −5.264393312251782, −4.965049155007946, −4.205971526609173, −3.396283577502008, −2.818925514880421, −2.079532454505125, −1.299665685481985, −0.4898067074925211,
0.4898067074925211, 1.299665685481985, 2.079532454505125, 2.818925514880421, 3.396283577502008, 4.205971526609173, 4.965049155007946, 5.264393312251782, 5.971844822688002, 6.248794451030042, 7.263769946195125, 7.412544286663916, 8.166960425121719, 8.705785076276377, 9.388214716413399, 10.02550692501579, 10.18158354496041, 10.91385561862521, 11.38488051570087, 11.78205123845034, 12.53806558087557, 12.97736127477390, 13.26584602953944, 14.00955253649703, 14.26253144566799