| L(s) = 1 | + 5.90e4·3-s + 2.57e7·5-s + 3.59e8·7-s + 3.48e9·9-s − 1.36e11·11-s − 3.92e11·13-s + 1.52e12·15-s − 1.39e13·17-s − 1.50e13·19-s + 2.12e13·21-s − 8.76e13·23-s + 1.86e14·25-s + 2.05e14·27-s − 2.21e15·29-s + 8.94e14·31-s − 8.04e15·33-s + 9.24e15·35-s − 3.48e16·37-s − 2.31e16·39-s + 1.42e17·41-s + 1.89e17·43-s + 8.98e16·45-s − 3.48e17·47-s − 4.29e17·49-s − 8.25e17·51-s + 1.82e18·53-s − 3.51e18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.17·5-s + 0.480·7-s + 0.333·9-s − 1.58·11-s − 0.789·13-s + 0.680·15-s − 1.68·17-s − 0.564·19-s + 0.277·21-s − 0.440·23-s + 0.391·25-s + 0.192·27-s − 0.977·29-s + 0.195·31-s − 0.914·33-s + 0.566·35-s − 1.19·37-s − 0.455·39-s + 1.65·41-s + 1.33·43-s + 0.393·45-s − 0.965·47-s − 0.769·49-s − 0.971·51-s + 1.43·53-s − 1.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
| good | 5 | \( 1 - 2.57e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 3.59e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.36e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.92e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.39e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.50e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 8.76e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.21e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 8.94e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.48e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.42e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.89e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.48e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.82e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.09e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.46e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.86e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.89e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.41e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 4.80e18T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.73e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.41e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.17e20T + 5.27e41T^{2} \) |
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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
−12.82543098112553329592445500146, −10.90497310331493345037732732350, −9.854499594534445510829455834797, −8.643426997106857078481767906273, −7.32978625181578211487271813738, −5.74983200609964490056042543756, −4.53427204154887230269740220040, −2.54075401135225115366094645312, −1.95093161606341506037033351908, 0,
1.95093161606341506037033351908, 2.54075401135225115366094645312, 4.53427204154887230269740220040, 5.74983200609964490056042543756, 7.32978625181578211487271813738, 8.643426997106857078481767906273, 9.854499594534445510829455834797, 10.90497310331493345037732732350, 12.82543098112553329592445500146
