| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 4·11-s + 15-s + 6·17-s + 4·21-s − 4·23-s + 25-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 4·35-s + 2·37-s − 10·41-s − 4·43-s − 45-s − 8·47-s + 9·49-s − 6·51-s − 2·53-s + 4·55-s − 4·59-s + 14·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s + 1.45·17-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s + 0.539·55-s − 0.520·59-s + 1.79·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.00509240175021, −15.45711302097043, −15.09608595669834, −14.21229217533429, −13.62176843654128, −13.02219806245832, −12.67356833289210, −12.13258841874703, −11.59857246911927, −10.96843758624327, −10.17813588703830, −9.903680239245969, −9.574409330416588, −8.417879071870238, −8.032386011424747, −7.414850819730763, −6.626512130844669, −6.313770906821267, −5.372653949081750, −5.180530121304308, −4.083198424388971, −3.440435137927237, −2.964878814249906, −1.997312698768280, −0.7587208522045789, 0,
0.7587208522045789, 1.997312698768280, 2.964878814249906, 3.440435137927237, 4.083198424388971, 5.180530121304308, 5.372653949081750, 6.313770906821267, 6.626512130844669, 7.414850819730763, 8.032386011424747, 8.417879071870238, 9.574409330416588, 9.903680239245969, 10.17813588703830, 10.96843758624327, 11.59857246911927, 12.13258841874703, 12.67356833289210, 13.02219806245832, 13.62176843654128, 14.21229217533429, 15.09608595669834, 15.45711302097043, 16.00509240175021
