| L(s) = 1 | + 3.25·3-s − 2.84·5-s − 1.37·7-s + 7.60·9-s + 11-s + 4.50·13-s − 9.25·15-s + 8.04·17-s + 19-s − 4.47·21-s − 1.93·23-s + 3.08·25-s + 14.9·27-s − 6.01·29-s − 2.25·31-s + 3.25·33-s + 3.90·35-s + 6.40·37-s + 14.6·39-s − 4.47·41-s + 2.91·43-s − 21.6·45-s + 11.0·47-s − 5.11·49-s + 26.1·51-s − 2.68·53-s − 2.84·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 1.27·5-s − 0.519·7-s + 2.53·9-s + 0.301·11-s + 1.24·13-s − 2.39·15-s + 1.95·17-s + 0.229·19-s − 0.976·21-s − 0.402·23-s + 0.617·25-s + 2.88·27-s − 1.11·29-s − 0.404·31-s + 0.566·33-s + 0.660·35-s + 1.05·37-s + 2.34·39-s − 0.699·41-s + 0.443·43-s − 3.22·45-s + 1.60·47-s − 0.730·49-s + 3.66·51-s − 0.369·53-s − 0.383·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543378030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543378030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 1.37T + 7T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 - 8.04T + 17T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 + 6.01T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 0.975T + 61T^{2} \) |
| 67 | \( 1 + 6.49T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 + 4.33T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 0.0256T + 97T^{2} \) |
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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
−9.908875719904010593119146525412, −9.226302158352172343371222394915, −8.432151892096774060964038370467, −7.73699452130216194178798245573, −7.31085690667651474651863719481, −5.94081056791129460047067522164, −4.23396990248103369912634333936, −3.58935168601552883051009090261, −3.05073021839015591164294384593, −1.39496532899414041389587703430,
1.39496532899414041389587703430, 3.05073021839015591164294384593, 3.58935168601552883051009090261, 4.23396990248103369912634333936, 5.94081056791129460047067522164, 7.31085690667651474651863719481, 7.73699452130216194178798245573, 8.432151892096774060964038370467, 9.226302158352172343371222394915, 9.908875719904010593119146525412
