| L(s) = 1 | + 9.01e4·3-s + 8.69e7·5-s + 2.82e9·7-s − 8.60e10·9-s + 1.78e12·11-s + 4.82e12·13-s + 7.83e12·15-s − 7.88e13·17-s − 8.73e14·19-s + 2.54e14·21-s − 6.19e15·23-s − 4.35e15·25-s − 1.62e16·27-s + 1.10e17·29-s − 5.17e16·31-s + 1.61e17·33-s + 2.45e17·35-s − 1.11e18·37-s + 4.34e17·39-s − 2.03e18·41-s − 8.20e17·43-s − 7.48e18·45-s − 2.81e19·47-s − 1.94e19·49-s − 7.10e18·51-s + 4.33e18·53-s + 1.55e20·55-s + ⋯ |
| L(s) = 1 | + 0.293·3-s + 0.796·5-s + 0.539·7-s − 0.913·9-s + 1.89·11-s + 0.746·13-s + 0.233·15-s − 0.557·17-s − 1.71·19-s + 0.158·21-s − 1.35·23-s − 0.365·25-s − 0.561·27-s + 1.68·29-s − 0.365·31-s + 0.555·33-s + 0.429·35-s − 1.03·37-s + 0.219·39-s − 0.576·41-s − 0.134·43-s − 0.727·45-s − 1.65·47-s − 0.709·49-s − 0.163·51-s + 0.0641·53-s + 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 9.01e4T + 9.41e10T^{2} \) |
| 5 | \( 1 - 8.69e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 2.82e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.78e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 4.82e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 7.88e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 8.73e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 6.19e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.10e17T + 4.31e33T^{2} \) |
| 31 | \( 1 + 5.17e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.11e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 2.03e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 8.20e17T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.81e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 4.33e18T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.42e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.86e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.37e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 3.62e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.26e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 2.95e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 2.85e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.88e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 3.46e22T + 4.96e45T^{2} \) |
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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
−10.06056512778378823389521418067, −8.836540621611525574696647855303, −8.377956808402018241908405128090, −6.53487087321146248701948950477, −6.04717385402584941763842702481, −4.53471984429598009783503923033, −3.53394230143831581156790536612, −2.09813697269962431362257876158, −1.48642705021552340978048410806, 0,
1.48642705021552340978048410806, 2.09813697269962431362257876158, 3.53394230143831581156790536612, 4.53471984429598009783503923033, 6.04717385402584941763842702481, 6.53487087321146248701948950477, 8.377956808402018241908405128090, 8.836540621611525574696647855303, 10.06056512778378823389521418067
