| L(s) = 1 | + 1.77e5·3-s + 3.07e6·5-s + 6.69e9·7-s + 3.13e10·9-s + 1.09e12·11-s + 9.54e10·13-s + 5.44e11·15-s − 2.80e14·17-s − 6.10e14·19-s + 1.18e15·21-s − 1.07e15·23-s − 1.19e16·25-s + 5.55e15·27-s − 8.47e16·29-s − 2.18e16·31-s + 1.93e17·33-s + 2.05e16·35-s − 1.71e18·37-s + 1.69e16·39-s + 4.07e18·41-s + 7.16e18·43-s + 9.64e16·45-s − 1.19e19·47-s + 1.74e19·49-s − 4.96e19·51-s − 5.50e19·53-s + 3.36e18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0281·5-s + 1.27·7-s + 0.333·9-s + 1.15·11-s + 0.0147·13-s + 0.0162·15-s − 1.98·17-s − 1.20·19-s + 0.738·21-s − 0.235·23-s − 0.999·25-s + 0.192·27-s − 1.28·29-s − 0.154·31-s + 0.668·33-s + 0.0360·35-s − 1.58·37-s + 0.00852·39-s + 1.15·41-s + 1.17·43-s + 0.00938·45-s − 0.704·47-s + 0.636·49-s − 1.14·51-s − 0.816·53-s + 0.0325·55-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 | \( 1 - 3.07e6T + 1.19e16T^{2} \) |
| 7 | \( 1 - 6.69e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.09e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 9.54e10T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.80e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 6.10e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 1.07e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 8.47e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 2.18e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.71e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.07e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 7.16e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.19e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 5.50e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.14e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.24e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 9.94e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.08e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.58e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 9.14e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 6.46e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.72e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 8.96e22T + 4.96e45T^{2} \) |
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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
−10.80241464053936174976705582877, −9.222743791041294020091541570090, −8.531586449057876222755009712736, −7.36230473926253667656207306181, −6.16924936115059175366816514620, −4.59492664176560647860682852197, −3.89274080708078817296303032547, −2.16427604459853005823344194071, −1.60979279909510487389042113185, 0,
1.60979279909510487389042113185, 2.16427604459853005823344194071, 3.89274080708078817296303032547, 4.59492664176560647860682852197, 6.16924936115059175366816514620, 7.36230473926253667656207306181, 8.531586449057876222755009712736, 9.222743791041294020091541570090, 10.80241464053936174976705582877
