| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 2·7-s + 1.73·8-s + 3.46·11-s + 13-s − 3.46·14-s − 5·16-s + 6.92·17-s + 2·19-s − 5.99·22-s − 6.92·23-s − 5·25-s − 1.73·26-s + 1.99·28-s − 6.92·29-s + 2·31-s + 5.19·32-s − 11.9·34-s + 2·37-s − 3.46·38-s − 6.92·41-s + 8·43-s + 3.46·44-s + 11.9·46-s + 10.3·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.755·7-s + 0.612·8-s + 1.04·11-s + 0.277·13-s − 0.925·14-s − 1.25·16-s + 1.68·17-s + 0.458·19-s − 1.27·22-s − 1.44·23-s − 25-s − 0.339·26-s + 0.377·28-s − 1.28·29-s + 0.359·31-s + 0.918·32-s − 2.05·34-s + 0.328·37-s − 0.561·38-s − 1.08·41-s + 1.21·43-s + 0.522·44-s + 1.76·46-s + 1.51·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6406829349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6406829349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−13.78077984813502739949344616735, −12.14597944474043241962403385795, −11.31185930034328724256934016158, −10.11191089374798319109999442731, −9.333019073369794308499960380212, −8.169496008074871911723438887652, −7.43939891686361858773967966201, −5.76356334736852262895132650365, −4.03630049463173406978464715852, −1.51141997204695526390325899979,
1.51141997204695526390325899979, 4.03630049463173406978464715852, 5.76356334736852262895132650365, 7.43939891686361858773967966201, 8.169496008074871911723438887652, 9.333019073369794308499960380212, 10.11191089374798319109999442731, 11.31185930034328724256934016158, 12.14597944474043241962403385795, 13.78077984813502739949344616735
