| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 11-s − 2·13-s − 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 6·29-s − 33-s − 35-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s + 45-s + 49-s − 2·51-s − 2·53-s + 55-s + 4·57-s − 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.174·33-s − 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.134·55-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569160937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569160937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.81377444878745, −15.25943453517136, −14.60071680849628, −14.24246853150291, −13.41060809188060, −13.07783274474702, −12.29079950836961, −12.08673331695323, −11.34955109915970, −10.62165595148809, −10.28452996465196, −9.624983068444776, −9.168288296743176, −8.390177542155349, −7.777189631945291, −7.001879702991683, −6.473232792414776, −6.004618238929894, −5.288734413746491, −4.640211357060898, −4.023295407016638, −3.098956202469107, −2.390941107076949, −1.498041251248678, −0.5556143593437719,
0.5556143593437719, 1.498041251248678, 2.390941107076949, 3.098956202469107, 4.023295407016638, 4.640211357060898, 5.288734413746491, 6.004618238929894, 6.473232792414776, 7.001879702991683, 7.777189631945291, 8.390177542155349, 9.168288296743176, 9.624983068444776, 10.28452996465196, 10.62165595148809, 11.34955109915970, 12.08673331695323, 12.29079950836961, 13.07783274474702, 13.41060809188060, 14.24246853150291, 14.60071680849628, 15.25943453517136, 15.81377444878745
