L(s) = 1 | − 1.77e5·3-s − 4.08e5·5-s + 9.39e9·7-s + 3.13e10·9-s − 8.83e11·11-s − 2.21e12·13-s + 7.23e10·15-s + 2.27e14·17-s + 6.53e13·19-s − 1.66e15·21-s − 5.34e15·23-s − 1.19e16·25-s − 5.55e15·27-s − 4.32e16·29-s − 1.98e17·31-s + 1.56e17·33-s − 3.83e15·35-s + 1.09e17·37-s + 3.91e17·39-s + 6.10e18·41-s − 2.13e18·43-s − 1.28e16·45-s − 1.58e19·47-s + 6.09e19·49-s − 4.03e19·51-s + 6.82e18·53-s + 3.60e17·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.00373·5-s + 1.79·7-s + 0.333·9-s − 0.933·11-s − 0.342·13-s + 0.00215·15-s + 1.61·17-s + 0.128·19-s − 1.03·21-s − 1.16·23-s − 0.999·25-s − 0.192·27-s − 0.658·29-s − 1.40·31-s + 0.539·33-s − 0.00671·35-s + 0.101·37-s + 0.197·39-s + 1.73·41-s − 0.349·43-s − 0.00124·45-s − 0.935·47-s + 2.22·49-s − 0.931·51-s + 0.101·53-s + 0.00349·55-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 | \( 1 + 4.08e5T + 1.19e16T^{2} \) |
| 7 | \( 1 - 9.39e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 8.83e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 2.21e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.27e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.53e13T + 2.57e29T^{2} \) |
| 23 | \( 1 + 5.34e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.32e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.98e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.09e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 6.10e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.13e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.58e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 6.82e18T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.82e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.11e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.00e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.75e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.77e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.85e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.00e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.15e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 9.73e22T + 4.96e45T^{2} \) |
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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
−10.81208224408548791797351949279, −9.714887635398139284310751516200, −7.998651354983664488642441834936, −7.59013713145671304485773152779, −5.70491188449638427267641030257, −5.11932667355313533777161189333, −3.90064361836643802895883080759, −2.20218897228025168740722839432, −1.28537192455792517765599342808, 0,
1.28537192455792517765599342808, 2.20218897228025168740722839432, 3.90064361836643802895883080759, 5.11932667355313533777161189333, 5.70491188449638427267641030257, 7.59013713145671304485773152779, 7.998651354983664488642441834936, 9.714887635398139284310751516200, 10.81208224408548791797351949279