L(s) = 1 | − 2.28·2-s + 0.445·3-s + 3.21·4-s + 5-s − 1.01·6-s + 7-s − 2.76·8-s − 2.80·9-s − 2.28·10-s + 0.882·11-s + 1.43·12-s − 0.264·13-s − 2.28·14-s + 0.445·15-s − 0.110·16-s + 6.80·17-s + 6.39·18-s + 2.33·19-s + 3.21·20-s + 0.445·21-s − 2.01·22-s + 23-s − 1.23·24-s + 25-s + 0.604·26-s − 2.58·27-s + 3.21·28-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 0.257·3-s + 1.60·4-s + 0.447·5-s − 0.415·6-s + 0.377·7-s − 0.977·8-s − 0.933·9-s − 0.721·10-s + 0.265·11-s + 0.413·12-s − 0.0734·13-s − 0.610·14-s + 0.115·15-s − 0.0277·16-s + 1.65·17-s + 1.50·18-s + 0.535·19-s + 0.718·20-s + 0.0973·21-s − 0.429·22-s + 0.208·23-s − 0.251·24-s + 0.200·25-s + 0.118·26-s − 0.497·27-s + 0.606·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8564458461\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8564458461\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 0.445T + 3T^{2} \) |
| 11 | \( 1 - 0.882T + 11T^{2} \) |
| 13 | \( 1 + 0.264T + 13T^{2} \) |
| 17 | \( 1 - 6.80T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 0.193T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.14606019168351663071044501286, −9.236399634639100667808905589958, −8.747291415173361259415469403714, −7.86809789352049850493222727858, −7.24456812774366124352991165365, −6.06377479872008585648928785903, −5.15464661815116682136896400328, −3.39957531117154418565337335007, −2.22630132083387521955893226267, −0.987964141753873532255665783945,
0.987964141753873532255665783945, 2.22630132083387521955893226267, 3.39957531117154418565337335007, 5.15464661815116682136896400328, 6.06377479872008585648928785903, 7.24456812774366124352991165365, 7.86809789352049850493222727858, 8.747291415173361259415469403714, 9.236399634639100667808905589958, 10.14606019168351663071044501286