| L(s) = 1 | + 9·3-s − 48.1·5-s − 75.0·7-s + 81·9-s + 714.·11-s − 1.11e3·13-s − 433.·15-s + 443.·17-s + 1.86e3·19-s − 675.·21-s − 835.·23-s − 809·25-s + 729·27-s − 5.01e3·29-s − 205.·31-s + 6.43e3·33-s + 3.61e3·35-s + 3.07e3·37-s − 1.00e4·39-s − 6.45e3·41-s − 1.03e4·43-s − 3.89e3·45-s + 2.55e4·47-s − 1.11e4·49-s + 3.98e3·51-s + 1.76e4·53-s − 3.44e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.860·5-s − 0.579·7-s + 0.333·9-s + 1.78·11-s − 1.83·13-s − 0.497·15-s + 0.371·17-s + 1.18·19-s − 0.334·21-s − 0.329·23-s − 0.258·25-s + 0.192·27-s − 1.10·29-s − 0.0384·31-s + 1.02·33-s + 0.498·35-s + 0.369·37-s − 1.05·39-s − 0.600·41-s − 0.849·43-s − 0.286·45-s + 1.68·47-s − 0.664·49-s + 0.214·51-s + 0.861·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.917048867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917048867\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| good | 5 | \( 1 + 48.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 75.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 714.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 443.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 835.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 205.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.55e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.76e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.67e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.33e3T + 8.58e9T^{2} \) |
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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
−9.524208065350712103828008161126, −8.864070981372068466266551128781, −7.57238673766024243085911033156, −7.31488383122149400692790236510, −6.19651701597898010940154817898, −4.90547593616152856921588515195, −3.88038093158440966920034198542, −3.25434566499798538101817064971, −1.95710430934103831708847517673, −0.61230139705382976111491286636,
0.61230139705382976111491286636, 1.95710430934103831708847517673, 3.25434566499798538101817064971, 3.88038093158440966920034198542, 4.90547593616152856921588515195, 6.19651701597898010940154817898, 7.31488383122149400692790236510, 7.57238673766024243085911033156, 8.864070981372068466266551128781, 9.524208065350712103828008161126
