| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 2·51-s − 6·53-s + 4·55-s + 8·59-s + 2·61-s − 2·65-s − 4·67-s + 4·71-s + 10·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 1.04·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.474·71-s + 1.17·73-s − 0.115·75-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
| \(L(1)\) |
\(\approx\) |
\(1.413335210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413335210\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.68026654536651, −15.00034293587327, −14.42026000583062, −13.70413802439138, −13.28344616870382, −12.70054836375831, −12.12178851420028, −11.73651463227078, −10.95660095589310, −10.70103031951806, −9.991905789881615, −9.594680468103642, −8.605063127315466, −8.143500841303859, −7.733726498379919, −6.933443226572415, −6.391770136096998, −5.744883660080102, −5.107494898720570, −4.571592006585672, −3.870198356932843, −3.022583509930152, −2.457080086085446, −1.291502846289273, −0.5392962175495003,
0.5392962175495003, 1.291502846289273, 2.457080086085446, 3.022583509930152, 3.870198356932843, 4.571592006585672, 5.107494898720570, 5.744883660080102, 6.391770136096998, 6.933443226572415, 7.733726498379919, 8.143500841303859, 8.605063127315466, 9.594680468103642, 9.991905789881615, 10.70103031951806, 10.95660095589310, 11.73651463227078, 12.12178851420028, 12.70054836375831, 13.28344616870382, 13.70413802439138, 14.42026000583062, 15.00034293587327, 15.68026654536651
