| L(s) = 1 | + 0.759·3-s − 2.04·7-s − 2.42·9-s − 1.34·11-s − 1.31·13-s − 4.08·17-s + 4.88·19-s − 1.55·21-s − 2.73·23-s − 4.11·27-s − 4.61·29-s − 7.15·31-s − 1.02·33-s + 8.64·37-s − 1.00·39-s − 10.0·41-s + 2.43·43-s + 7.57·47-s − 2.82·49-s − 3.10·51-s + 0.621·53-s + 3.70·57-s + 11.3·59-s + 0.647·61-s + 4.94·63-s + 10.9·67-s − 2.07·69-s + ⋯ |
| L(s) = 1 | + 0.438·3-s − 0.771·7-s − 0.807·9-s − 0.406·11-s − 0.366·13-s − 0.991·17-s + 1.12·19-s − 0.338·21-s − 0.570·23-s − 0.792·27-s − 0.857·29-s − 1.28·31-s − 0.178·33-s + 1.42·37-s − 0.160·39-s − 1.57·41-s + 0.371·43-s + 1.10·47-s − 0.404·49-s − 0.434·51-s + 0.0853·53-s + 0.491·57-s + 1.47·59-s + 0.0829·61-s + 0.623·63-s + 1.33·67-s − 0.250·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155767547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155767547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.759T + 3T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 7.57T + 47T^{2} \) |
| 53 | \( 1 - 0.621T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.647T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−7.58739800509300057095614737679, −7.11939175523143404720633366757, −6.24987246293265583230008738865, −5.63437744016942496287008129535, −5.01993468648257382799441591484, −3.97188991231271282911922558272, −3.37718013009656530428217607918, −2.61527180156462722484376292351, −1.96227003610402243767824356632, −0.46536893337921514522341194860,
0.46536893337921514522341194860, 1.96227003610402243767824356632, 2.61527180156462722484376292351, 3.37718013009656530428217607918, 3.97188991231271282911922558272, 5.01993468648257382799441591484, 5.63437744016942496287008129535, 6.24987246293265583230008738865, 7.11939175523143404720633366757, 7.58739800509300057095614737679
