| L(s) = 1 | + 13.9·3-s + 25·5-s + 49·7-s − 48.2·9-s − 473.·11-s + 22.8·13-s + 348.·15-s − 1.78e3·17-s − 2.95e3·19-s + 683.·21-s + 3.33e3·23-s + 625·25-s − 4.06e3·27-s − 4.71e3·29-s − 1.59e3·31-s − 6.60e3·33-s + 1.22e3·35-s + 7.78e3·37-s + 319.·39-s − 5.78e3·41-s − 4.47e3·43-s − 1.20e3·45-s + 2.95e4·47-s + 2.40e3·49-s − 2.48e4·51-s − 2.43e4·53-s − 1.18e4·55-s + ⋯ |
| L(s) = 1 | + 0.895·3-s + 0.447·5-s + 0.377·7-s − 0.198·9-s − 1.17·11-s + 0.0375·13-s + 0.400·15-s − 1.49·17-s − 1.87·19-s + 0.338·21-s + 1.31·23-s + 0.200·25-s − 1.07·27-s − 1.04·29-s − 0.297·31-s − 1.05·33-s + 0.169·35-s + 0.935·37-s + 0.0336·39-s − 0.537·41-s − 0.369·43-s − 0.0887·45-s + 1.95·47-s + 0.142·49-s − 1.33·51-s − 1.18·53-s − 0.527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 13.9T + 243T^{2} \) |
| 11 | \( 1 + 473.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 22.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.95e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.47e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.50e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e4T + 8.58e9T^{2} \) |
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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
−10.69878128479496992391839404731, −9.320074672698247077801301368512, −8.668158274588073224379254174996, −7.83464341061390383836449758844, −6.63400957874845551085782197836, −5.40567821544252933332267483474, −4.24315584067897383890586769303, −2.74245815212572737521802825782, −2.00852005888322678381302962654, 0,
2.00852005888322678381302962654, 2.74245815212572737521802825782, 4.24315584067897383890586769303, 5.40567821544252933332267483474, 6.63400957874845551085782197836, 7.83464341061390383836449758844, 8.668158274588073224379254174996, 9.320074672698247077801301368512, 10.69878128479496992391839404731
