| L(s) = 1 | − 3-s + 1.37·5-s + 3.37·7-s + 9-s − 1.37·11-s − 2·13-s − 1.37·15-s + 1.37·17-s − 19-s − 3.37·21-s − 8.74·23-s − 3.11·25-s − 27-s − 2.74·29-s − 6.74·31-s + 1.37·33-s + 4.62·35-s − 4.74·37-s + 2·39-s − 3.37·43-s + 1.37·45-s − 13.3·47-s + 4.37·49-s − 1.37·51-s + 2.74·53-s − 1.88·55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.613·5-s + 1.27·7-s + 0.333·9-s − 0.413·11-s − 0.554·13-s − 0.354·15-s + 0.332·17-s − 0.229·19-s − 0.735·21-s − 1.82·23-s − 0.623·25-s − 0.192·27-s − 0.509·29-s − 1.21·31-s + 0.238·33-s + 0.782·35-s − 0.780·37-s + 0.320·39-s − 0.514·43-s + 0.204·45-s − 1.95·47-s + 0.624·49-s − 0.192·51-s + 0.376·53-s − 0.253·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−7.998086233245645609863997773574, −7.58509638452749418940194772715, −6.60441208735645178971606909332, −5.73373079441141996735671466391, −5.26715529210277998792176626758, −4.52005807405867325512969520152, −3.57910270161836211003849754816, −2.11953453969528317688254481511, −1.66740198243381287478997505626, 0,
1.66740198243381287478997505626, 2.11953453969528317688254481511, 3.57910270161836211003849754816, 4.52005807405867325512969520152, 5.26715529210277998792176626758, 5.73373079441141996735671466391, 6.60441208735645178971606909332, 7.58509638452749418940194772715, 7.998086233245645609863997773574
