| L(s) = 1 | + 2-s + 4-s − 1.73·5-s + 1.26·7-s + 8-s − 1.73·10-s − 1.26·11-s + 1.26·14-s + 16-s − 5.19·17-s + 4.73·19-s − 1.73·20-s − 1.26·22-s + 8.19·23-s − 2.00·25-s + 1.26·28-s + 3·29-s + 9.46·31-s + 32-s − 5.19·34-s − 2.19·35-s + 3·37-s + 4.73·38-s − 1.73·40-s − 6.46·41-s − 4.19·43-s − 1.26·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.774·5-s + 0.479·7-s + 0.353·8-s − 0.547·10-s − 0.382·11-s + 0.338·14-s + 0.250·16-s − 1.26·17-s + 1.08·19-s − 0.387·20-s − 0.270·22-s + 1.70·23-s − 0.400·25-s + 0.239·28-s + 0.557·29-s + 1.69·31-s + 0.176·32-s − 0.891·34-s − 0.371·35-s + 0.493·37-s + 0.767·38-s − 0.273·40-s − 1.00·41-s − 0.639·43-s − 0.191·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659821151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659821151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.394954276680925924230407291964, −8.054524933309666890774180561198, −6.99733113537378081800058731230, −6.64322499263362940144347190670, −5.36069361030688311095974606106, −4.87090390824712768481786189131, −4.08189789689010512488697510855, −3.17280689697497580960092811354, −2.31979612613075247521586249959, −0.909448153655508590737341082519,
0.909448153655508590737341082519, 2.31979612613075247521586249959, 3.17280689697497580960092811354, 4.08189789689010512488697510855, 4.87090390824712768481786189131, 5.36069361030688311095974606106, 6.64322499263362940144347190670, 6.99733113537378081800058731230, 8.054524933309666890774180561198, 8.394954276680925924230407291964
