| L(s) = 1 | + 3-s − 2.73·5-s + 7-s + 9-s − 4·11-s + 5.46·13-s − 2.73·15-s − 6.73·17-s + 19-s + 21-s + 6.92·23-s + 2.46·25-s + 27-s − 6.73·29-s − 2·31-s − 4·33-s − 2.73·35-s + 8.92·37-s + 5.46·39-s + 4.92·41-s − 8.92·43-s − 2.73·45-s − 10.1·47-s + 49-s − 6.73·51-s − 9.66·53-s + 10.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.22·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s + 1.51·13-s − 0.705·15-s − 1.63·17-s + 0.229·19-s + 0.218·21-s + 1.44·23-s + 0.492·25-s + 0.192·27-s − 1.25·29-s − 0.359·31-s − 0.696·33-s − 0.461·35-s + 1.46·37-s + 0.874·39-s + 0.769·41-s − 1.36·43-s − 0.407·45-s − 1.48·47-s + 0.142·49-s − 0.942·51-s − 1.32·53-s + 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.233094778261918892145358667709, −7.73591949671411004845472382152, −7.01509303503386221066223596200, −6.10424659975622265851487578233, −4.99746464758065943525636689581, −4.32522413119396594845933725526, −3.51066016760584286453364321721, −2.74266292618532147556381359504, −1.50974378961662278794505562502, 0,
1.50974378961662278794505562502, 2.74266292618532147556381359504, 3.51066016760584286453364321721, 4.32522413119396594845933725526, 4.99746464758065943525636689581, 6.10424659975622265851487578233, 7.01509303503386221066223596200, 7.73591949671411004845472382152, 8.233094778261918892145358667709
