| L(s) = 1 | + 0.435·2-s − 3-s − 1.81·4-s + 4.22·5-s − 0.435·6-s − 2.90·7-s − 1.65·8-s + 9-s + 1.83·10-s + 0.559·11-s + 1.81·12-s + 1.50·13-s − 1.26·14-s − 4.22·15-s + 2.89·16-s + 0.435·18-s + 6.57·19-s − 7.64·20-s + 2.90·21-s + 0.243·22-s − 1.98·23-s + 1.65·24-s + 12.8·25-s + 0.653·26-s − 27-s + 5.26·28-s − 2.24·29-s + ⋯ |
| L(s) = 1 | + 0.307·2-s − 0.577·3-s − 0.905·4-s + 1.88·5-s − 0.177·6-s − 1.09·7-s − 0.586·8-s + 0.333·9-s + 0.581·10-s + 0.168·11-s + 0.522·12-s + 0.416·13-s − 0.338·14-s − 1.08·15-s + 0.724·16-s + 0.102·18-s + 1.50·19-s − 1.70·20-s + 0.634·21-s + 0.0519·22-s − 0.414·23-s + 0.338·24-s + 2.56·25-s + 0.128·26-s − 0.192·27-s + 0.995·28-s − 0.417·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555235006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555235006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.435T + 2T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 - 0.559T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 4.90T + 37T^{2} \) |
| 41 | \( 1 - 7.01T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 0.354T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 3.56T + 67T^{2} \) |
| 71 | \( 1 + 3.72T + 71T^{2} \) |
| 73 | \( 1 - 8.61T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 9.16T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−9.949799769081406093949085094599, −9.437331285737346815343839738353, −8.964143105983958687636406208362, −7.40368607529912051342115198694, −6.21822001204779953367999058224, −5.84394940043633749347225417404, −5.12314091907152533287379051248, −3.85011561472312963805413108806, −2.67302605501544954268123419977, −1.05112400659151548298716561096,
1.05112400659151548298716561096, 2.67302605501544954268123419977, 3.85011561472312963805413108806, 5.12314091907152533287379051248, 5.84394940043633749347225417404, 6.21822001204779953367999058224, 7.40368607529912051342115198694, 8.964143105983958687636406208362, 9.437331285737346815343839738353, 9.949799769081406093949085094599
