| L(s) = 1 | − 14.5·3-s − 44.8·5-s − 31.8·9-s − 634.·11-s − 20.5·13-s + 651.·15-s − 1.05e3·17-s + 65.1·19-s − 4.05e3·23-s − 1.11e3·25-s + 3.99e3·27-s − 6.58e3·29-s + 1.08e3·31-s + 9.22e3·33-s + 685.·37-s + 299.·39-s − 1.36e4·41-s − 1.60e4·43-s + 1.42e3·45-s − 1.36e4·47-s + 1.52e4·51-s − 1.60e3·53-s + 2.84e4·55-s − 947.·57-s − 7.52e3·59-s + 2.08e4·61-s + 922.·65-s + ⋯ |
| L(s) = 1 | − 0.932·3-s − 0.801·5-s − 0.131·9-s − 1.58·11-s − 0.0337·13-s + 0.747·15-s − 0.881·17-s + 0.0414·19-s − 1.59·23-s − 0.357·25-s + 1.05·27-s − 1.45·29-s + 0.203·31-s + 1.47·33-s + 0.0822·37-s + 0.0314·39-s − 1.26·41-s − 1.32·43-s + 0.105·45-s − 0.898·47-s + 0.821·51-s − 0.0786·53-s + 1.26·55-s − 0.0386·57-s − 0.281·59-s + 0.718·61-s + 0.0270·65-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
| \(L(3)\) |
\(\approx\) |
\(0.07751682560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07751682560\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 14.5T + 243T^{2} \) |
| 5 | \( 1 + 44.8T + 3.12e3T^{2} \) |
| 11 | \( 1 + 634.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 20.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 65.1T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 685.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.36e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.52e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.87e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.76e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.68e5T + 8.58e9T^{2} \) |
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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
−10.68810448039111769367516312461, −9.815452227311524449752770579652, −8.384155633103902449831717376715, −7.78560069936173743072722487430, −6.62194772427121394381556963164, −5.59549426833076439385662795697, −4.79114170484227011096562398546, −3.55197902741125787262590235530, −2.13365320993537865592372630542, −0.14460181661546123593830035972,
0.14460181661546123593830035972, 2.13365320993537865592372630542, 3.55197902741125787262590235530, 4.79114170484227011096562398546, 5.59549426833076439385662795697, 6.62194772427121394381556963164, 7.78560069936173743072722487430, 8.384155633103902449831717376715, 9.815452227311524449752770579652, 10.68810448039111769367516312461
