| L(s) = 1 | − 5·5-s − 29.9·7-s − 56.7·11-s + 30.3·13-s − 99.8·17-s − 81.8·19-s − 137.·23-s + 25·25-s + 49.0·29-s + 3.50·31-s + 149.·35-s − 289.·37-s + 199.·41-s − 516.·43-s + 109.·47-s + 552.·49-s − 283.·53-s + 283.·55-s + 185.·59-s − 235.·61-s − 151.·65-s − 341.·67-s − 1.13e3·71-s + 1.11e3·73-s + 1.69e3·77-s − 200.·79-s − 205.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.61·7-s − 1.55·11-s + 0.648·13-s − 1.42·17-s − 0.988·19-s − 1.24·23-s + 0.200·25-s + 0.314·29-s + 0.0203·31-s + 0.722·35-s − 1.28·37-s + 0.761·41-s − 1.83·43-s + 0.340·47-s + 1.61·49-s − 0.734·53-s + 0.696·55-s + 0.409·59-s − 0.495·61-s − 0.289·65-s − 0.622·67-s − 1.89·71-s + 1.78·73-s + 2.51·77-s − 0.285·79-s − 0.271·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1565050829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1565050829\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 + 56.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.50T + 2.97e4T^{2} \) |
| 37 | \( 1 + 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 199.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 516.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 235.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 200.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 205.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 350.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 923.T + 9.12e5T^{2} \) |
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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
−8.890981278349851774338571385524, −8.362306126386360698817546207776, −7.39188895209236413369437681333, −6.52245087306904214347838057913, −5.99906850419420630500811954812, −4.82674720312887052551552862003, −3.87748834217502946943797421438, −3.03036276122505137001619475961, −2.10303245303226050743036577134, −0.17505438123884286030761215195,
0.17505438123884286030761215195, 2.10303245303226050743036577134, 3.03036276122505137001619475961, 3.87748834217502946943797421438, 4.82674720312887052551552862003, 5.99906850419420630500811954812, 6.52245087306904214347838057913, 7.39188895209236413369437681333, 8.362306126386360698817546207776, 8.890981278349851774338571385524
