| L(s) = 1 | − 18.3·3-s − 84.3·5-s − 216.·7-s + 92.8·9-s − 380.·11-s + 1.05e3·13-s + 1.54e3·15-s + 801.·17-s + 288.·19-s + 3.97e3·21-s − 334.·23-s + 3.98e3·25-s + 2.75e3·27-s + 841·29-s + 3.28e3·31-s + 6.97e3·33-s + 1.82e4·35-s − 1.11e4·37-s − 1.93e4·39-s − 887.·41-s − 1.32e4·43-s − 7.83e3·45-s + 2.33e3·47-s + 3.02e4·49-s − 1.46e4·51-s + 1.85e3·53-s + 3.20e4·55-s + ⋯ |
| L(s) = 1 | − 1.17·3-s − 1.50·5-s − 1.67·7-s + 0.382·9-s − 0.948·11-s + 1.73·13-s + 1.77·15-s + 0.672·17-s + 0.183·19-s + 1.96·21-s − 0.131·23-s + 1.27·25-s + 0.726·27-s + 0.185·29-s + 0.613·31-s + 1.11·33-s + 2.52·35-s − 1.33·37-s − 2.04·39-s − 0.0824·41-s − 1.09·43-s − 0.576·45-s + 0.154·47-s + 1.79·49-s − 0.790·51-s + 0.0904·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 29 | \( 1 - 841T \) |
| good | 3 | \( 1 + 18.3T + 243T^{2} \) |
| 5 | \( 1 + 84.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 216.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 380.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 801.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 288.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 334.T + 6.43e6T^{2} \) |
| 31 | \( 1 - 3.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 887.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.10e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.62e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.33e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.46e4T + 8.58e9T^{2} \) |
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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
−10.14460445440216015176008509421, −8.785562468568085225061021376626, −7.917931578344731163622422487136, −6.80425911765349056117453514232, −6.12673063319064178869979012887, −5.13096456649192261627481895974, −3.76746192345694361023648427297, −3.15741222159101438052906970724, −0.78706283178381512172470686841, 0,
0.78706283178381512172470686841, 3.15741222159101438052906970724, 3.76746192345694361023648427297, 5.13096456649192261627481895974, 6.12673063319064178869979012887, 6.80425911765349056117453514232, 7.917931578344731163622422487136, 8.785562468568085225061021376626, 10.14460445440216015176008509421
