L(s) = 1 | + 0.795·3-s + 5-s + 2.08·7-s − 2.36·9-s + 3.33·11-s + 3.89·13-s + 0.795·15-s − 6.62·17-s + 1.82·19-s + 1.65·21-s + 5.82·23-s + 25-s − 4.27·27-s + 2.07·29-s + 5.43·31-s + 2.65·33-s + 2.08·35-s + 0.0603·37-s + 3.09·39-s + 3.93·41-s + 3.95·43-s − 2.36·45-s + 1.86·47-s − 2.65·49-s − 5.26·51-s − 5.15·53-s + 3.33·55-s + ⋯ |
L(s) = 1 | + 0.459·3-s + 0.447·5-s + 0.787·7-s − 0.788·9-s + 1.00·11-s + 1.07·13-s + 0.205·15-s − 1.60·17-s + 0.417·19-s + 0.361·21-s + 1.21·23-s + 0.200·25-s − 0.821·27-s + 0.385·29-s + 0.976·31-s + 0.462·33-s + 0.352·35-s + 0.00992·37-s + 0.496·39-s + 0.615·41-s + 0.602·43-s − 0.352·45-s + 0.271·47-s − 0.379·49-s − 0.737·51-s − 0.708·53-s + 0.449·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.238005001\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.238005001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 0.795T + 3T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 - 2.07T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 0.0603T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 + 5.15T + 53T^{2} \) |
| 59 | \( 1 + 4.27T + 59T^{2} \) |
| 61 | \( 1 + 6.28T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 - 4.30T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 8.10T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−8.076604812654119410825014629781, −7.07435495331878533676985271644, −6.40452635271566009474821406427, −5.89554820897161990392299455733, −4.92421524691759978187104930452, −4.34457382305491904693732066864, −3.43051645240995144185440793664, −2.65658903082699757330158330774, −1.79161840389080714118537948875, −0.917194649431653710372475840887,
0.917194649431653710372475840887, 1.79161840389080714118537948875, 2.65658903082699757330158330774, 3.43051645240995144185440793664, 4.34457382305491904693732066864, 4.92421524691759978187104930452, 5.89554820897161990392299455733, 6.40452635271566009474821406427, 7.07435495331878533676985271644, 8.076604812654119410825014629781