| L(s) = 1 | − 1.56e3·2-s − 5.94e6·4-s − 1.13e8·5-s + 7.85e9·7-s + 2.24e10·8-s + 1.77e11·10-s − 1.03e12·11-s + 8.08e12·13-s − 1.22e13·14-s + 1.48e13·16-s + 1.38e14·17-s − 1.41e14·19-s + 6.75e14·20-s + 1.61e15·22-s − 4.80e15·23-s + 9.90e14·25-s − 1.26e16·26-s − 4.66e16·28-s + 1.45e16·29-s − 8.10e16·31-s − 2.11e17·32-s − 2.16e17·34-s − 8.92e17·35-s + 1.19e18·37-s + 2.21e17·38-s − 2.54e18·40-s − 6.63e18·41-s + ⋯ |
| L(s) = 1 | − 0.539·2-s − 0.708·4-s − 1.04·5-s + 1.50·7-s + 0.922·8-s + 0.561·10-s − 1.08·11-s + 1.25·13-s − 0.810·14-s + 0.210·16-s + 0.981·17-s − 0.279·19-s + 0.737·20-s + 0.588·22-s − 1.05·23-s + 0.0831·25-s − 0.675·26-s − 1.06·28-s + 0.221·29-s − 0.573·31-s − 1.03·32-s − 0.529·34-s − 1.56·35-s + 1.10·37-s + 0.150·38-s − 0.960·40-s − 1.88·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.56e3T + 8.38e6T^{2} \) |
| 5 | \( 1 + 1.13e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 7.85e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.03e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 8.08e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.38e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 1.41e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 4.80e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.45e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 8.10e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.19e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 6.63e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.06e19T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.16e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.56e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 4.19e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 7.45e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.29e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.37e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.03e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 3.75e20T + 4.42e43T^{2} \) |
| 83 | \( 1 - 9.99e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 5.36e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.11e22T + 4.96e45T^{2} \) |
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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
−14.92902322066841086829643137371, −13.49439744827782343828957043602, −11.67531161131143678855108464763, −10.40523970032618676532142563323, −8.330296178940767330501082868492, −7.888735371192775187747329419950, −5.16094898286293167284693698437, −3.85694554063887107580338314261, −1.43785345789369896871991193766, 0,
1.43785345789369896871991193766, 3.85694554063887107580338314261, 5.16094898286293167284693698437, 7.888735371192775187747329419950, 8.330296178940767330501082868492, 10.40523970032618676532142563323, 11.67531161131143678855108464763, 13.49439744827782343828957043602, 14.92902322066841086829643137371
