| L(s) = 1 | + 3-s + 2·7-s + 9-s − 0.449·11-s − 13-s + 2·17-s + 6.89·19-s + 2·21-s − 4.89·23-s + 27-s − 2·29-s − 2.89·31-s − 0.449·33-s + 10.8·37-s − 39-s + 3.55·41-s + 7.79·43-s + 5.34·47-s − 3·49-s + 2·51-s − 2.89·53-s + 6.89·57-s − 4.44·59-s + 4·61-s + 2·63-s − 7.79·67-s − 4.89·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 0.333·9-s − 0.135·11-s − 0.277·13-s + 0.485·17-s + 1.58·19-s + 0.436·21-s − 1.02·23-s + 0.192·27-s − 0.371·29-s − 0.520·31-s − 0.0782·33-s + 1.79·37-s − 0.160·39-s + 0.554·41-s + 1.18·43-s + 0.780·47-s − 0.428·49-s + 0.280·51-s − 0.398·53-s + 0.913·57-s − 0.579·59-s + 0.512·61-s + 0.251·63-s − 0.952·67-s − 0.589·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.746808283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.746808283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.449T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 3.55T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 2.89T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 7.79T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.79T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.342608120974980421214100330711, −7.58809920685543825856145957447, −7.48533595767222954074859680859, −6.16495810210354878584257391135, −5.47555562541101384672595262741, −4.64164918300943443064683146200, −3.84298289956904700987091730660, −2.92519433242924122133408971411, −2.03806162977200506277993751950, −0.974867019229906380358326718164,
0.974867019229906380358326718164, 2.03806162977200506277993751950, 2.92519433242924122133408971411, 3.84298289956904700987091730660, 4.64164918300943443064683146200, 5.47555562541101384672595262741, 6.16495810210354878584257391135, 7.48533595767222954074859680859, 7.58809920685543825856145957447, 8.342608120974980421214100330711
