| L(s) = 1 | − 0.464·3-s + 1.08·5-s − 2.78·9-s + 11-s + 0.499·13-s − 0.501·15-s + 1.84·17-s − 7.25·19-s + 2.10·23-s − 3.83·25-s + 2.68·27-s + 4.75·29-s + 6.51·31-s − 0.464·33-s − 2.96·37-s − 0.231·39-s − 1.40·41-s + 10.7·43-s − 3.01·45-s − 0.772·47-s − 0.856·51-s − 11.7·53-s + 1.08·55-s + 3.36·57-s + 12.5·59-s + 7.46·61-s + 0.540·65-s + ⋯ |
| L(s) = 1 | − 0.267·3-s + 0.483·5-s − 0.928·9-s + 0.301·11-s + 0.138·13-s − 0.129·15-s + 0.447·17-s − 1.66·19-s + 0.439·23-s − 0.766·25-s + 0.516·27-s + 0.883·29-s + 1.16·31-s − 0.0807·33-s − 0.487·37-s − 0.0371·39-s − 0.218·41-s + 1.64·43-s − 0.448·45-s − 0.112·47-s − 0.119·51-s − 1.60·53-s + 0.145·55-s + 0.445·57-s + 1.63·59-s + 0.955·61-s + 0.0670·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628584758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628584758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.464T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 13 | \( 1 - 0.499T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 0.772T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 + 0.683T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.470458602206181043574665175364, −7.76481110959793672795599236899, −6.62572582575079656672161070068, −6.26027699470763569173108696236, −5.51491920422178214659224947028, −4.71320253980733903828306724839, −3.83639286455553471743513057012, −2.81954593792755307497531022680, −2.02323178360170883977133558040, −0.71943161245967802969149004110,
0.71943161245967802969149004110, 2.02323178360170883977133558040, 2.81954593792755307497531022680, 3.83639286455553471743513057012, 4.71320253980733903828306724839, 5.51491920422178214659224947028, 6.26027699470763569173108696236, 6.62572582575079656672161070068, 7.76481110959793672795599236899, 8.470458602206181043574665175364
