L(s) = 1 | + 168.·5-s − 13.9·7-s + 1.50e3·11-s + 744. i·13-s − 2.96e3i·17-s + 6.31e3i·19-s − 9.99e3i·23-s + 1.29e4·25-s + 2.03e4·29-s + 3.92e4·31-s − 2.36e3·35-s + 8.92e4i·37-s + 2.00e4i·41-s − 5.54e4i·43-s − 1.08e5i·47-s + ⋯ |
L(s) = 1 | + 1.35·5-s − 0.0407·7-s + 1.12·11-s + 0.339i·13-s − 0.603i·17-s + 0.920i·19-s − 0.821i·23-s + 0.827·25-s + 0.836·29-s + 1.31·31-s − 0.0551·35-s + 1.76i·37-s + 0.291i·41-s − 0.697i·43-s − 1.04i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.506409262\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.506409262\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 168.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 13.9T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.50e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 744. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 6.31e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 9.99e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.03e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.92e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.92e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 2.00e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 5.54e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.08e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.43e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.10e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.20e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.03e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.39e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.79e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.85e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.10e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.99e5T + 8.32e11T^{2} \) |
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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
−9.892192753829730262045516310371, −9.019273645596810326204593186900, −8.204509335352286111135662821492, −6.71650862369209125768428592523, −6.35416902504998031270932123469, −5.26233767520322358044089081624, −4.24706675212439633543129293603, −2.92349048835765994868067162282, −1.85331888642928464372135155639, −0.941771162754981966908150071459,
0.856473617539958709941734299817, 1.81099895829681310920971194009, 2.87053729259096003994177316467, 4.14253903666426963082614576935, 5.28312226724768396862092713838, 6.16800603948146362440727692295, 6.79404813213604721110485301422, 8.059782756397518033024623773162, 9.137558349643538793134806120540, 9.602078718801628255255342498503