L(s) = 1 | + 1.09·2-s + 0.493·3-s − 0.794·4-s + 0.542·6-s − 7-s − 3.06·8-s − 2.75·9-s − 1.70·11-s − 0.392·12-s + 4.14·13-s − 1.09·14-s − 1.78·16-s − 3.30·17-s − 3.02·18-s + 2.28·19-s − 0.493·21-s − 1.87·22-s + 23-s − 1.51·24-s + 4.55·26-s − 2.84·27-s + 0.794·28-s + 3.22·29-s + 5.89·31-s + 4.18·32-s − 0.843·33-s − 3.63·34-s + ⋯ |
L(s) = 1 | + 0.776·2-s + 0.285·3-s − 0.397·4-s + 0.221·6-s − 0.377·7-s − 1.08·8-s − 0.918·9-s − 0.514·11-s − 0.113·12-s + 1.15·13-s − 0.293·14-s − 0.445·16-s − 0.801·17-s − 0.713·18-s + 0.523·19-s − 0.107·21-s − 0.399·22-s + 0.208·23-s − 0.309·24-s + 0.893·26-s − 0.546·27-s + 0.150·28-s + 0.599·29-s + 1.05·31-s + 0.739·32-s − 0.146·33-s − 0.622·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.950888406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.950888406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 - 0.493T + 3T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 8.57T + 83T^{2} \) |
| 89 | \( 1 + 3.30T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.445462889961555700057910086065, −7.933317550395369680237519218856, −6.65234774251283461686762596910, −6.16358639415010581883664353707, −5.37865583490600608664497926490, −4.71091712700877659395925654061, −3.74607054018903462666174522722, −3.16398486642984790577909858379, −2.35763087165908383970584449379, −0.68338040569004458944626418174,
0.68338040569004458944626418174, 2.35763087165908383970584449379, 3.16398486642984790577909858379, 3.74607054018903462666174522722, 4.71091712700877659395925654061, 5.37865583490600608664497926490, 6.16358639415010581883664353707, 6.65234774251283461686762596910, 7.933317550395369680237519218856, 8.445462889961555700057910086065