L(s) = 1 | + 3.89i·3-s + 6.27·7-s − 6.14·9-s − 11.1i·11-s − 15.9i·13-s − 23.9·17-s + 7.55i·19-s + 24.4i·21-s + (−18.3 + 13.8i)23-s + 11.1i·27-s − 10.3·29-s + 53.0·31-s + 43.4·33-s − 26.7·37-s + 62.1·39-s + ⋯ |
L(s) = 1 | + 1.29i·3-s + 0.896·7-s − 0.682·9-s − 1.01i·11-s − 1.22i·13-s − 1.40·17-s + 0.397i·19-s + 1.16i·21-s + (−0.799 + 0.600i)23-s + 0.411i·27-s − 0.356·29-s + 1.71·31-s + 1.31·33-s − 0.723·37-s + 1.59·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.053762089\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.053762089\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (18.3 - 13.8i)T \) |
good | 3 | \( 1 - 3.89iT - 9T^{2} \) |
| 7 | \( 1 - 6.27T + 49T^{2} \) |
| 11 | \( 1 + 11.1iT - 121T^{2} \) |
| 13 | \( 1 + 15.9iT - 169T^{2} \) |
| 17 | \( 1 + 23.9T + 289T^{2} \) |
| 19 | \( 1 - 7.55iT - 361T^{2} \) |
| 29 | \( 1 + 10.3T + 841T^{2} \) |
| 31 | \( 1 - 53.0T + 961T^{2} \) |
| 37 | \( 1 + 26.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 11.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 56.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 92.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 78.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 113. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 55.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 59.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 9.67iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 75.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 141.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 21.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.6T + 9.40e3T^{2} \) |
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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
−8.714268160763431156185377554628, −8.369859179783681264199030977113, −7.50027853728856579354671469110, −6.29592152558333786222618906564, −5.48571645425558732287265636207, −4.85316911172418543012388335400, −4.00802591725455423146876360846, −3.29654491164263899178765546730, −2.11824861810422587409551632952, −0.59326085676320765683054807046,
0.948165461170461981889839983835, 2.04385729832616212318000505177, 2.36053554134757671587296772954, 4.29713751790085049094754406655, 4.56938070308778691423984017603, 5.89967530241927343045713468621, 6.78703272979987744785494629911, 7.07759880038936750829719706354, 7.979611481582977821647433770313, 8.608935182506003765110582964828