| L(s) = 1 | + 34.6·2-s + 690.·4-s − 1.85e3·5-s − 1.15e4·7-s + 6.17e3·8-s − 6.43e4·10-s − 7.53e4·11-s − 1.25e5·13-s − 4.00e5·14-s − 1.39e5·16-s − 1.22e5·17-s + 4.24e5·19-s − 1.28e6·20-s − 2.61e6·22-s + 8.84e5·23-s + 1.48e6·25-s − 4.35e6·26-s − 7.96e6·28-s − 2.04e6·29-s − 6.30e6·31-s − 7.98e6·32-s − 4.25e6·34-s + 2.14e7·35-s − 1.74e7·37-s + 1.47e7·38-s − 1.14e7·40-s − 2.62e6·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.34·4-s − 1.32·5-s − 1.81·7-s + 0.533·8-s − 2.03·10-s − 1.55·11-s − 1.21·13-s − 2.78·14-s − 0.530·16-s − 0.356·17-s + 0.748·19-s − 1.78·20-s − 2.37·22-s + 0.659·23-s + 0.762·25-s − 1.86·26-s − 2.44·28-s − 0.538·29-s − 1.22·31-s − 1.34·32-s − 0.546·34-s + 2.41·35-s − 1.53·37-s + 1.14·38-s − 0.708·40-s − 0.145·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.001464743513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001464743513\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 - 34.6T + 512T^{2} \) |
| 5 | \( 1 + 1.85e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.15e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.53e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.25e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.22e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.24e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.84e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.30e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.74e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.62e6T + 3.27e14T^{2} \) |
| 47 | \( 1 - 2.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.08e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.13e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.50e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.92e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.76e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.20e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.39e9T + 7.60e17T^{2} \) |
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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
−9.996612763542361123875298994893, −8.894995774243324405630698373907, −7.34856017770359209545578344385, −7.10290246108043696254321424197, −5.72667006463600300120950024637, −4.98914693145896431969783296206, −3.88584007902042303297486530725, −3.17779639330384277107922631776, −2.52582209618854504725896579378, −0.01141791983058075667040858189,
0.01141791983058075667040858189, 2.52582209618854504725896579378, 3.17779639330384277107922631776, 3.88584007902042303297486530725, 4.98914693145896431969783296206, 5.72667006463600300120950024637, 7.10290246108043696254321424197, 7.34856017770359209545578344385, 8.894995774243324405630698373907, 9.996612763542361123875298994893
