| L(s) = 1 | − 2.10·3-s + 5-s + 2.10·7-s + 1.42·9-s + 5.62·11-s + 13-s − 2.10·15-s + 17-s + 4·19-s − 4.42·21-s − 1.62·23-s − 4·25-s + 3.31·27-s + 7.83·29-s + 9.62·31-s − 11.8·33-s + 2.10·35-s + 0.421·37-s − 2.10·39-s − 5.83·41-s + 0.475·43-s + 1.42·45-s − 4.68·47-s − 2.57·49-s − 2.10·51-s + 4.57·53-s + 5.62·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 0.447·5-s + 0.794·7-s + 0.473·9-s + 1.69·11-s + 0.277·13-s − 0.542·15-s + 0.242·17-s + 0.917·19-s − 0.964·21-s − 0.339·23-s − 0.800·25-s + 0.638·27-s + 1.45·29-s + 1.72·31-s − 2.05·33-s + 0.355·35-s + 0.0693·37-s − 0.336·39-s − 0.910·41-s + 0.0725·43-s + 0.211·45-s − 0.682·47-s − 0.368·49-s − 0.294·51-s + 0.628·53-s + 0.758·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735400558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735400558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 - 9.62T + 31T^{2} \) |
| 37 | \( 1 - 0.421T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 - 0.475T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.558484183984894354490584417404, −7.950187260775300107320703052100, −6.76525990078994895013931477756, −6.38204159646396665696384159308, −5.66872736735170351506491023433, −4.87423169360005331640311501532, −4.22730660039605320190231745102, −3.08361693116337505833122516687, −1.64507650596156503579249083779, −0.927278825092054858832407298121,
0.927278825092054858832407298121, 1.64507650596156503579249083779, 3.08361693116337505833122516687, 4.22730660039605320190231745102, 4.87423169360005331640311501532, 5.66872736735170351506491023433, 6.38204159646396665696384159308, 6.76525990078994895013931477756, 7.950187260775300107320703052100, 8.558484183984894354490584417404
