| L(s) = 1 | − 0.509·3-s + 1.39·5-s + 1.69·7-s − 2.74·9-s + 11-s − 6.15·13-s − 0.712·15-s − 4.10·17-s − 1.66·19-s − 0.862·21-s + 8.76·23-s − 3.04·25-s + 2.92·27-s + 4.00·29-s + 8.47·31-s − 0.509·33-s + 2.37·35-s − 3.12·37-s + 3.13·39-s + 2.86·41-s + 6.54·43-s − 3.83·45-s + 1.13·47-s − 4.12·49-s + 2.09·51-s + 4.32·53-s + 1.39·55-s + ⋯ |
| L(s) = 1 | − 0.293·3-s + 0.625·5-s + 0.640·7-s − 0.913·9-s + 0.301·11-s − 1.70·13-s − 0.184·15-s − 0.995·17-s − 0.381·19-s − 0.188·21-s + 1.82·23-s − 0.608·25-s + 0.562·27-s + 0.744·29-s + 1.52·31-s − 0.0886·33-s + 0.400·35-s − 0.513·37-s + 0.501·39-s + 0.447·41-s + 0.997·43-s − 0.571·45-s + 0.165·47-s − 0.589·49-s + 0.292·51-s + 0.593·53-s + 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 0.509T + 3T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 8.76T + 23T^{2} \) |
| 29 | \( 1 - 4.00T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−7.69256584424718217231039906920, −6.93994878787288493017717724440, −6.30273898733628041190367863948, −5.53503538509621322090414725733, −4.81161260732272682202119827799, −4.39612532169148099303644066182, −2.85497873079794063918878585326, −2.49366764102211056897184830067, −1.33707548869434473570820449483, 0,
1.33707548869434473570820449483, 2.49366764102211056897184830067, 2.85497873079794063918878585326, 4.39612532169148099303644066182, 4.81161260732272682202119827799, 5.53503538509621322090414725733, 6.30273898733628041190367863948, 6.93994878787288493017717724440, 7.69256584424718217231039906920
