L(s) = 1 | − 0.552·2-s + 3-s − 1.69·4-s + 1.59·5-s − 0.552·6-s + 2.04·8-s + 9-s − 0.878·10-s − 11-s − 1.69·12-s + 2.87·13-s + 1.59·15-s + 2.26·16-s + 4.83·17-s − 0.552·18-s − 1.14·19-s − 2.69·20-s + 0.552·22-s − 3.65·23-s + 2.04·24-s − 2.47·25-s − 1.59·26-s + 27-s − 0.325·29-s − 0.878·30-s + 6.45·31-s − 5.33·32-s + ⋯ |
L(s) = 1 | − 0.390·2-s + 0.577·3-s − 0.847·4-s + 0.711·5-s − 0.225·6-s + 0.721·8-s + 0.333·9-s − 0.277·10-s − 0.301·11-s − 0.489·12-s + 0.798·13-s + 0.410·15-s + 0.565·16-s + 1.17·17-s − 0.130·18-s − 0.262·19-s − 0.602·20-s + 0.117·22-s − 0.761·23-s + 0.416·24-s − 0.494·25-s − 0.311·26-s + 0.192·27-s − 0.0605·29-s − 0.160·30-s + 1.15·31-s − 0.942·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.689001528\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.689001528\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.552T + 2T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.325T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 - 0.305T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 - 8.40T + 83T^{2} \) |
| 89 | \( 1 + 5.74T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−9.516766089369598989096554722498, −8.443196126004444539658553456914, −8.207522036497753271804602808147, −7.20592520257999139379962265238, −6.03654251744295865224157914489, −5.36224523272015526715219831680, −4.26180071192347383021294275443, −3.45626527878571792642022303482, −2.18125986026744260407396493520, −1.00114930610388602003518001870,
1.00114930610388602003518001870, 2.18125986026744260407396493520, 3.45626527878571792642022303482, 4.26180071192347383021294275443, 5.36224523272015526715219831680, 6.03654251744295865224157914489, 7.20592520257999139379962265238, 8.207522036497753271804602808147, 8.443196126004444539658553456914, 9.516766089369598989096554722498