| L(s) = 1 | − 18.4·3-s + 25·5-s − 55.3·7-s + 97·9-s + 479.·11-s − 506·13-s − 460.·15-s − 1.83e3·17-s + 2.06e3·19-s + 1.02e3·21-s + 1.89e3·23-s + 625·25-s + 2.69e3·27-s + 4.53e3·29-s + 3.65e3·31-s − 8.84e3·33-s − 1.38e3·35-s − 338·37-s + 9.33e3·39-s − 6.33e3·41-s − 1.81e4·43-s + 2.42e3·45-s − 4.03e3·47-s − 1.37e4·49-s + 3.38e4·51-s + 1.54e4·53-s + 1.19e4·55-s + ⋯ |
| L(s) = 1 | − 1.18·3-s + 0.447·5-s − 0.426·7-s + 0.399·9-s + 1.19·11-s − 0.830·13-s − 0.528·15-s − 1.54·17-s + 1.31·19-s + 0.504·21-s + 0.748·23-s + 0.200·25-s + 0.710·27-s + 1.00·29-s + 0.682·31-s − 1.41·33-s − 0.190·35-s − 0.0405·37-s + 0.982·39-s − 0.588·41-s − 1.49·43-s + 0.178·45-s − 0.266·47-s − 0.817·49-s + 1.82·51-s + 0.757·53-s + 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 25T \) |
| good | 3 | \( 1 + 18.4T + 243T^{2} \) |
| 7 | \( 1 + 55.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 479.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 506T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.89e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 338T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.03e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.54e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.30e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.06e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.99e4T + 8.58e9T^{2} \) |
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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
−10.35212908401473964995243833909, −9.564908514980904409388048327247, −8.617288808949196002667787519238, −6.90958867311967858645830288513, −6.52860424707105337957494176307, −5.36930919915339972189008497464, −4.50096014775731499497515377709, −2.90273577245660092594108140969, −1.27929016129050089799264296738, 0,
1.27929016129050089799264296738, 2.90273577245660092594108140969, 4.50096014775731499497515377709, 5.36930919915339972189008497464, 6.52860424707105337957494176307, 6.90958867311967858645830288513, 8.617288808949196002667787519238, 9.564908514980904409388048327247, 10.35212908401473964995243833909
