| L(s) = 1 | + 1.77e5·3-s − 1.57e8·5-s − 8.81e9·7-s + 3.13e10·9-s + 4.03e10·11-s + 7.60e12·13-s − 2.78e13·15-s − 1.79e14·17-s + 5.45e14·19-s − 1.56e15·21-s + 8.25e15·23-s + 1.27e16·25-s + 5.55e15·27-s + 8.83e16·29-s + 1.96e16·31-s + 7.14e15·33-s + 1.38e18·35-s + 2.53e17·37-s + 1.34e18·39-s − 5.07e17·41-s − 1.05e19·43-s − 4.93e18·45-s + 1.49e19·47-s + 5.02e19·49-s − 3.17e19·51-s − 6.83e19·53-s − 6.33e18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.43·5-s − 1.68·7-s + 0.333·9-s + 0.0426·11-s + 1.17·13-s − 0.830·15-s − 1.26·17-s + 1.07·19-s − 0.972·21-s + 1.80·23-s + 1.07·25-s + 0.192·27-s + 1.34·29-s + 0.138·31-s + 0.0246·33-s + 2.42·35-s + 0.234·37-s + 0.679·39-s − 0.144·41-s − 1.73·43-s − 0.479·45-s + 0.884·47-s + 1.83·49-s − 0.732·51-s − 1.01·53-s − 0.0613·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.77e5T \) |
| good | 5 | \( 1 + 1.57e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 8.81e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 4.03e10T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.60e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.79e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.45e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 8.25e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 8.83e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.96e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 2.53e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 5.07e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.05e19T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.49e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 6.83e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 3.48e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.54e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 9.99e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.37e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.48e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 4.32e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.35e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.27e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.23e23T + 4.96e45T^{2} \) |
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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
−10.67501670974779945821567762904, −9.295414319762156293164132663645, −8.493945182708751832649064003447, −7.21347990298182914851860643302, −6.40814748166212655761565439644, −4.55750884094484700016016603173, −3.42464966227836082179832819908, −2.98350393826152419354258064565, −1.02964955938552513238433792969, 0,
1.02964955938552513238433792969, 2.98350393826152419354258064565, 3.42464966227836082179832819908, 4.55750884094484700016016603173, 6.40814748166212655761565439644, 7.21347990298182914851860643302, 8.493945182708751832649064003447, 9.295414319762156293164132663645, 10.67501670974779945821567762904
