| L(s) = 1 | + 2.30·2-s + 3.30·4-s + 5-s − 7-s + 3.00·8-s + 2.30·10-s + 11-s + 3.60·13-s − 2.30·14-s + 0.302·16-s + 4·17-s + 3·19-s + 3.30·20-s + 2.30·22-s + 2·23-s − 4·25-s + 8.30·26-s − 3.30·28-s − 5.60·29-s − 2·31-s − 5.30·32-s + 9.21·34-s − 35-s − 8.21·37-s + 6.90·38-s + 3.00·40-s − 7.21·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.728·10-s + 0.301·11-s + 1.00·13-s − 0.615·14-s + 0.0756·16-s + 0.970·17-s + 0.688·19-s + 0.738·20-s + 0.490·22-s + 0.417·23-s − 0.800·25-s + 1.62·26-s − 0.624·28-s − 1.04·29-s − 0.359·31-s − 0.937·32-s + 1.57·34-s − 0.169·35-s − 1.34·37-s + 1.12·38-s + 0.474·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.903749989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.903749989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8.21T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 - 2.39T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−10.70796566540872969329755229258, −9.735986155121920974463703062957, −8.789195515370035215501904505887, −7.47994158733684855720487267118, −6.55917500509980125677895294161, −5.75796334656372880573643349782, −5.13572645756635745181213101822, −3.78340489930993696855598114187, −3.24905057990207045718136250013, −1.75222576013048732290404434874,
1.75222576013048732290404434874, 3.24905057990207045718136250013, 3.78340489930993696855598114187, 5.13572645756635745181213101822, 5.75796334656372880573643349782, 6.55917500509980125677895294161, 7.47994158733684855720487267118, 8.789195515370035215501904505887, 9.735986155121920974463703062957, 10.70796566540872969329755229258
