| L(s) = 1 | + 2.33·2-s + 1.99·3-s + 3.45·4-s − 5-s + 4.65·6-s + 3.39·8-s + 0.977·9-s − 2.33·10-s − 2.05·11-s + 6.88·12-s − 5.30·13-s − 1.99·15-s + 1.01·16-s − 3.88·17-s + 2.28·18-s + 0.886·19-s − 3.45·20-s − 4.80·22-s − 0.744·23-s + 6.76·24-s + 25-s − 12.3·26-s − 4.03·27-s − 8.44·29-s − 4.65·30-s − 0.531·31-s − 4.41·32-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.15·3-s + 1.72·4-s − 0.447·5-s + 1.90·6-s + 1.19·8-s + 0.325·9-s − 0.738·10-s − 0.620·11-s + 1.98·12-s − 1.47·13-s − 0.514·15-s + 0.253·16-s − 0.942·17-s + 0.538·18-s + 0.203·19-s − 0.771·20-s − 1.02·22-s − 0.155·23-s + 1.38·24-s + 0.200·25-s − 2.42·26-s − 0.776·27-s − 1.56·29-s − 0.850·30-s − 0.0954·31-s − 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 - 1.99T + 3T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 19 | \( 1 - 0.886T + 19T^{2} \) |
| 23 | \( 1 + 0.744T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 + 0.531T + 31T^{2} \) |
| 41 | \( 1 - 0.590T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 9.13T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 0.554T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 + 8.27T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.36611311310052798272682636122, −6.74980863782982774488970906010, −5.72693435401672836221627880441, −5.20768871799627497111396713399, −4.43764619480320626904733300350, −3.86959202658448190597851109333, −3.14996693694181324623935063778, −2.46151217692854989745516457082, −2.04185273418166769110700667443, 0,
2.04185273418166769110700667443, 2.46151217692854989745516457082, 3.14996693694181324623935063778, 3.86959202658448190597851109333, 4.43764619480320626904733300350, 5.20768871799627497111396713399, 5.72693435401672836221627880441, 6.74980863782982774488970906010, 7.36611311310052798272682636122
