| L(s) = 1 | + 9.99·2-s − 27.4·3-s + 67.8·4-s − 274.·6-s − 29.7·7-s + 358.·8-s + 511.·9-s − 179.·11-s − 1.86e3·12-s + 169·13-s − 297.·14-s + 1.40e3·16-s + 1.31e3·17-s + 5.10e3·18-s − 1.55e3·19-s + 818.·21-s − 1.79e3·22-s − 4.87e3·23-s − 9.83e3·24-s + 1.68e3·26-s − 7.35e3·27-s − 2.02e3·28-s + 7.31e3·29-s − 1.61e3·31-s + 2.60e3·32-s + 4.94e3·33-s + 1.31e4·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.76·3-s + 2.11·4-s − 3.11·6-s − 0.229·7-s + 1.97·8-s + 2.10·9-s − 0.448·11-s − 3.73·12-s + 0.277·13-s − 0.406·14-s + 1.37·16-s + 1.10·17-s + 3.71·18-s − 0.987·19-s + 0.404·21-s − 0.792·22-s − 1.92·23-s − 3.48·24-s + 0.489·26-s − 1.94·27-s − 0.487·28-s + 1.61·29-s − 0.301·31-s + 0.448·32-s + 0.790·33-s + 1.95·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 9.99T + 32T^{2} \) |
| 3 | \( 1 + 27.4T + 243T^{2} \) |
| 7 | \( 1 + 29.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 179.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.31e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.53e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5T + 8.58e9T^{2} \) |
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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
−10.65631294363772409106747934471, −10.08776028973501134731107608627, −7.924410314379776063164047280117, −6.58963953086218031858774070993, −6.18321977004902932338517666304, −5.26335684211966765168537419413, −4.54559335424532040159288143203, −3.41927332372214496445173391946, −1.72668040597491627609012376965, 0,
1.72668040597491627609012376965, 3.41927332372214496445173391946, 4.54559335424532040159288143203, 5.26335684211966765168537419413, 6.18321977004902932338517666304, 6.58963953086218031858774070993, 7.924410314379776063164047280117, 10.08776028973501134731107608627, 10.65631294363772409106747934471
