| L(s) = 1 | + 1.42·3-s − 3.22·5-s − 0.429·7-s − 0.956·9-s − 11-s + 4.88·13-s − 4.61·15-s + 3.70·17-s − 19-s − 0.614·21-s + 3.32·23-s + 5.40·25-s − 5.65·27-s − 2.75·29-s − 0.699·31-s − 1.42·33-s + 1.38·35-s − 1.61·37-s + 6.97·39-s − 8.99·41-s − 7.40·43-s + 3.08·45-s + 10.9·47-s − 6.81·49-s + 5.30·51-s + 0.452·53-s + 3.22·55-s + ⋯ |
| L(s) = 1 | + 0.825·3-s − 1.44·5-s − 0.162·7-s − 0.318·9-s − 0.301·11-s + 1.35·13-s − 1.19·15-s + 0.899·17-s − 0.229·19-s − 0.134·21-s + 0.692·23-s + 1.08·25-s − 1.08·27-s − 0.511·29-s − 0.125·31-s − 0.248·33-s + 0.234·35-s − 0.265·37-s + 1.11·39-s − 1.40·41-s − 1.12·43-s + 0.459·45-s + 1.59·47-s − 0.973·49-s + 0.742·51-s + 0.0621·53-s + 0.435·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 + 0.429T + 7T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 0.699T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.452T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 + 7.62T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 8.64T + 83T^{2} \) |
| 89 | \( 1 - 3.89T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.268908911698127851980396541868, −7.72538331325372940902321081062, −6.99443440721236372809967486617, −6.01547549143989926023810877619, −5.12736021939213026695364150501, −4.03722284797253118274272769439, −3.45875577748403795169004308534, −2.91155060690244138106957142700, −1.46150558430403383828690775447, 0,
1.46150558430403383828690775447, 2.91155060690244138106957142700, 3.45875577748403795169004308534, 4.03722284797253118274272769439, 5.12736021939213026695364150501, 6.01547549143989926023810877619, 6.99443440721236372809967486617, 7.72538331325372940902321081062, 8.268908911698127851980396541868
