| L(s) = 1 | + 3-s − 2·4-s + 1.73·7-s + 9-s − 2·12-s + 4·16-s − 3.46·17-s + 5.19·19-s + 1.73·21-s − 6·23-s + 27-s − 3.46·28-s − 6.92·29-s + 31-s − 2·36-s + 5·37-s − 3.46·41-s + 10.3·43-s + 12·47-s + 4·48-s − 4·49-s − 3.46·51-s − 6·53-s + 5.19·57-s + 12.1·61-s + 1.73·63-s − 8·64-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.654·7-s + 0.333·9-s − 0.577·12-s + 16-s − 0.840·17-s + 1.19·19-s + 0.377·21-s − 1.25·23-s + 0.192·27-s − 0.654·28-s − 1.28·29-s + 0.179·31-s − 0.333·36-s + 0.821·37-s − 0.541·41-s + 1.58·43-s + 1.75·47-s + 0.577·48-s − 0.571·49-s − 0.485·51-s − 0.824·53-s + 0.688·57-s + 1.55·61-s + 0.218·63-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021619383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021619383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−7.70675399911308994916913563422, −7.43563381746346935311325609793, −6.26749395255807075850422212455, −5.54950452968540898834852899590, −4.88504411183406661485358194193, −4.12936962423859347972747161874, −3.67925688564252219474166957945, −2.61428926486940176230141930942, −1.74944808436051574548016877352, −0.68543110405201829306145238655,
0.68543110405201829306145238655, 1.74944808436051574548016877352, 2.61428926486940176230141930942, 3.67925688564252219474166957945, 4.12936962423859347972747161874, 4.88504411183406661485358194193, 5.54950452968540898834852899590, 6.26749395255807075850422212455, 7.43563381746346935311325609793, 7.70675399911308994916913563422
