| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 6·13-s − 15-s + 6·17-s + 4·19-s + 8·23-s + 25-s + 27-s + 10·29-s + 4·31-s − 4·33-s − 6·37-s − 6·39-s − 6·41-s − 4·43-s − 45-s − 12·47-s + 6·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 2·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-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 + 0.718·31-s − 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162871107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162871107\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.33485740989688, −14.94532276473920, −14.43366378413432, −13.88765088791533, −13.32345974703862, −12.73493552234680, −12.11483407339370, −11.91648218207932, −11.07173962135358, −10.26267791934422, −9.988875208882367, −9.582684534095771, −8.503504510350801, −8.329297905271932, −7.565341686663925, −7.187673857018493, −6.661856274628163, −5.497565361462662, −4.957997292083325, −4.760297125575112, −3.540156533823400, −2.935290006531299, −2.682667259198023, −1.499566579414323, −0.5816266994947348,
0.5816266994947348, 1.499566579414323, 2.682667259198023, 2.935290006531299, 3.540156533823400, 4.760297125575112, 4.957997292083325, 5.497565361462662, 6.661856274628163, 7.187673857018493, 7.565341686663925, 8.329297905271932, 8.503504510350801, 9.582684534095771, 9.988875208882367, 10.26267791934422, 11.07173962135358, 11.91648218207932, 12.11483407339370, 12.73493552234680, 13.32345974703862, 13.88765088791533, 14.43366378413432, 14.94532276473920, 15.33485740989688
