| L(s) = 1 | + 2.82·3-s + 0.233·5-s − 7-s + 4.99·9-s + 11-s − 13-s + 0.659·15-s − 1.37·17-s − 2.67·19-s − 2.82·21-s − 3.07·23-s − 4.94·25-s + 5.64·27-s + 9.22·29-s + 6.01·31-s + 2.82·33-s − 0.233·35-s − 4.26·37-s − 2.82·39-s + 8.07·41-s + 6.43·43-s + 1.16·45-s + 11.0·47-s + 49-s − 3.89·51-s + 3.87·53-s + 0.233·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.104·5-s − 0.377·7-s + 1.66·9-s + 0.301·11-s − 0.277·13-s + 0.170·15-s − 0.333·17-s − 0.614·19-s − 0.617·21-s − 0.641·23-s − 0.989·25-s + 1.08·27-s + 1.71·29-s + 1.08·31-s + 0.492·33-s − 0.0394·35-s − 0.700·37-s − 0.452·39-s + 1.26·41-s + 0.981·43-s + 0.173·45-s + 1.61·47-s + 0.142·49-s − 0.545·51-s + 0.532·53-s + 0.0314·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.851951616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.851951616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.233T + 5T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 - 9.22T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 - 8.07T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 0.633T + 71T^{2} \) |
| 73 | \( 1 - 6.61T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−7.958862784506032282838922084672, −7.32979104666789545444956490673, −6.54830370734084552187227191202, −5.94314580396471500102335728627, −4.75331540603470013021221139983, −4.06856131152809673171384879342, −3.50931694238581275965121154683, −2.44510121976464074044270484972, −2.25989157905654449193559247134, −0.896102107191924041388679186407,
0.896102107191924041388679186407, 2.25989157905654449193559247134, 2.44510121976464074044270484972, 3.50931694238581275965121154683, 4.06856131152809673171384879342, 4.75331540603470013021221139983, 5.94314580396471500102335728627, 6.54830370734084552187227191202, 7.32979104666789545444956490673, 7.958862784506032282838922084672
