| L(s) = 1 | − 20.8·3-s − 90.4·5-s + 49·7-s + 193.·9-s − 552.·11-s + 593.·13-s + 1.88e3·15-s − 1.42e3·17-s + 318.·19-s − 1.02e3·21-s − 659.·23-s + 5.05e3·25-s + 1.03e3·27-s + 8.18e3·29-s + 9.59e3·31-s + 1.15e4·33-s − 4.43e3·35-s − 5.18e3·37-s − 1.23e4·39-s − 2.19e3·41-s + 7.45e3·43-s − 1.75e4·45-s + 1.95e4·47-s + 2.40e3·49-s + 2.97e4·51-s − 3.65e4·53-s + 4.99e4·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 1.61·5-s + 0.377·7-s + 0.796·9-s − 1.37·11-s + 0.973·13-s + 2.16·15-s − 1.19·17-s + 0.202·19-s − 0.506·21-s − 0.260·23-s + 1.61·25-s + 0.273·27-s + 1.80·29-s + 1.79·31-s + 1.84·33-s − 0.611·35-s − 0.622·37-s − 1.30·39-s − 0.203·41-s + 0.615·43-s − 1.28·45-s + 1.29·47-s + 0.142·49-s + 1.59·51-s − 1.78·53-s + 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 49T \) |
| good | 3 | \( 1 + 20.8T + 243T^{2} \) |
| 5 | \( 1 + 90.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 552.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 593.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 318.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 659.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.19e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.45e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.03e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.20e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
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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
−10.41035406791707009002423015958, −8.596074367887953312614244560960, −8.044896740201202461083124530445, −6.99172832880619937529163499338, −6.06723946955564904170353445435, −4.88001293633614855858320821168, −4.30923809049796453636003053518, −2.86800405709774728429459422844, −0.886765024171149552750762745212, 0,
0.886765024171149552750762745212, 2.86800405709774728429459422844, 4.30923809049796453636003053518, 4.88001293633614855858320821168, 6.06723946955564904170353445435, 6.99172832880619937529163499338, 8.044896740201202461083124530445, 8.596074367887953312614244560960, 10.41035406791707009002423015958
