| L(s) = 1 | + 0.374·3-s − 2.61·5-s + 4.69·7-s − 2.85·9-s − 6.32·11-s + 4.57·13-s − 0.979·15-s + 0.437·17-s − 7.21·19-s + 1.76·21-s + 1.10·23-s + 1.83·25-s − 2.19·27-s + 0.181·29-s + 2.73·31-s − 2.37·33-s − 12.2·35-s − 9.21·37-s + 1.71·39-s − 6.90·41-s − 6.29·43-s + 7.47·45-s − 4.12·47-s + 15.0·49-s + 0.163·51-s − 0.881·53-s + 16.5·55-s + ⋯ |
| L(s) = 1 | + 0.216·3-s − 1.16·5-s + 1.77·7-s − 0.953·9-s − 1.90·11-s + 1.26·13-s − 0.253·15-s + 0.106·17-s − 1.65·19-s + 0.384·21-s + 0.231·23-s + 0.366·25-s − 0.422·27-s + 0.0336·29-s + 0.490·31-s − 0.413·33-s − 2.07·35-s − 1.51·37-s + 0.274·39-s − 1.07·41-s − 0.959·43-s + 1.11·45-s − 0.601·47-s + 2.14·49-s + 0.0229·51-s − 0.121·53-s + 2.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 0.374T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 0.437T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 - 0.181T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 0.881T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 1.85T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 0.195T + 97T^{2} \) |
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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
−8.633856653405875898404795538859, −8.243543897366249570353017934003, −8.087896405287564738579309217149, −6.95461960793331959629277211255, −5.64017743077406851593936175866, −4.95939784933351619353894834177, −4.07201623747804523757795484522, −2.99552604846120172756766724170, −1.82756722322965835551001378987, 0,
1.82756722322965835551001378987, 2.99552604846120172756766724170, 4.07201623747804523757795484522, 4.95939784933351619353894834177, 5.64017743077406851593936175866, 6.95461960793331959629277211255, 8.087896405287564738579309217149, 8.243543897366249570353017934003, 8.633856653405875898404795538859
