| L(s) = 1 | + 3-s + 3.50·5-s − 4.71·7-s + 9-s + 2.19·11-s + 6.58·13-s + 3.50·15-s + 3.39·17-s + 5.90·19-s − 4.71·21-s − 23-s + 7.26·25-s + 27-s + 29-s − 5.12·31-s + 2.19·33-s − 16.4·35-s − 7.87·37-s + 6.58·39-s + 2.36·41-s + 3.79·43-s + 3.50·45-s − 10.9·47-s + 15.1·49-s + 3.39·51-s + 11.7·53-s + 7.69·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.56·5-s − 1.78·7-s + 0.333·9-s + 0.662·11-s + 1.82·13-s + 0.904·15-s + 0.823·17-s + 1.35·19-s − 1.02·21-s − 0.208·23-s + 1.45·25-s + 0.192·27-s + 0.185·29-s − 0.921·31-s + 0.382·33-s − 2.78·35-s − 1.29·37-s + 1.05·39-s + 0.368·41-s + 0.579·43-s + 0.522·45-s − 1.59·47-s + 2.16·49-s + 0.475·51-s + 1.61·53-s + 1.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.664245141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.664245141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 0.680T + 59T^{2} \) |
| 61 | \( 1 - 0.133T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 - 6.86T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.85663942698146969453455285678, −6.79642509710796636123154650557, −6.57702180601621964378124723586, −5.70565891462656790704299278666, −5.46579425834185707619422149560, −3.90965559887221485437108473704, −3.43080239424614821312787843080, −2.82411600329728501598403246049, −1.72725789033980710477070277915, −0.994015105558835996732493664474,
0.994015105558835996732493664474, 1.72725789033980710477070277915, 2.82411600329728501598403246049, 3.43080239424614821312787843080, 3.90965559887221485437108473704, 5.46579425834185707619422149560, 5.70565891462656790704299278666, 6.57702180601621964378124723586, 6.79642509710796636123154650557, 7.85663942698146969453455285678
