| L(s) = 1 | + 1.73·3-s − 5-s − 5.19·11-s − 1.73·15-s − 5·17-s + 1.73·19-s + 1.73·23-s − 4·25-s − 5.19·27-s + 8·29-s − 8.66·31-s − 9·33-s − 5·37-s − 4·41-s − 6.92·43-s + 8.66·47-s − 8.66·51-s − 53-s + 5.19·55-s + 2.99·57-s + 1.73·59-s − 11·61-s + 12.1·67-s + 2.99·69-s + 13.8·71-s − 15·73-s − 6.92·75-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.447·5-s − 1.56·11-s − 0.447·15-s − 1.21·17-s + 0.397·19-s + 0.361·23-s − 0.800·25-s − 1.00·27-s + 1.48·29-s − 1.55·31-s − 1.56·33-s − 0.821·37-s − 0.624·41-s − 1.05·43-s + 1.26·47-s − 1.21·51-s − 0.137·53-s + 0.700·55-s + 0.397·57-s + 0.225·59-s − 1.40·61-s + 1.48·67-s + 0.361·69-s + 1.64·71-s − 1.75·73-s − 0.800·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.832443402605427668192012600181, −8.293127084651543003213691605111, −7.62473822803488943409818211173, −6.85795088417944299202644685591, −5.62796939912276000558408700129, −4.81553158730289397814414536046, −3.71459672137746628721843940476, −2.87381173586581413683708153789, −2.04633501668482838765722432535, 0,
2.04633501668482838765722432535, 2.87381173586581413683708153789, 3.71459672137746628721843940476, 4.81553158730289397814414536046, 5.62796939912276000558408700129, 6.85795088417944299202644685591, 7.62473822803488943409818211173, 8.293127084651543003213691605111, 8.832443402605427668192012600181
