| L(s) = 1 | + 11.2·3-s + 26.0·5-s − 115.·9-s − 434.·11-s + 737.·13-s + 294.·15-s − 1.41e3·17-s + 1.73e3·19-s − 929.·23-s − 2.44e3·25-s − 4.04e3·27-s − 1.63e3·29-s − 1.87e3·31-s − 4.90e3·33-s − 7.93e3·37-s + 8.32e3·39-s + 6.32e3·41-s − 2.00e4·43-s − 3.01e3·45-s + 1.02e4·47-s − 1.59e4·51-s + 1.19e4·53-s − 1.13e4·55-s + 1.96e4·57-s − 2.11e4·59-s − 350.·61-s + 1.92e4·65-s + ⋯ |
| L(s) = 1 | + 0.724·3-s + 0.466·5-s − 0.475·9-s − 1.08·11-s + 1.20·13-s + 0.337·15-s − 1.18·17-s + 1.10·19-s − 0.366·23-s − 0.782·25-s − 1.06·27-s − 0.360·29-s − 0.349·31-s − 0.784·33-s − 0.952·37-s + 0.876·39-s + 0.587·41-s − 1.65·43-s − 0.221·45-s + 0.677·47-s − 0.858·51-s + 0.585·53-s − 0.505·55-s + 0.799·57-s − 0.792·59-s − 0.0120·61-s + 0.564·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 11.2T + 243T^{2} \) |
| 5 | \( 1 - 26.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 737.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 929.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.32e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 350.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 843.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.55e5T + 8.58e9T^{2} \) |
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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
−9.946880051700913737230462839085, −8.989554371098312220082266209368, −8.326100009559699880875946978551, −7.38024613391251120957931499633, −6.09377209201433841384606663189, −5.27762740669882427671233497537, −3.81051818523758741035362465922, −2.77978009142229358274167049172, −1.73207824445426739516513972053, 0,
1.73207824445426739516513972053, 2.77978009142229358274167049172, 3.81051818523758741035362465922, 5.27762740669882427671233497537, 6.09377209201433841384606663189, 7.38024613391251120957931499633, 8.326100009559699880875946978551, 8.989554371098312220082266209368, 9.946880051700913737230462839085
