L(s) = 1 | + 81i·3-s − 473. i·5-s − 574.·7-s − 6.56e3·9-s − 7.27e4i·11-s + 1.48e4i·13-s + 3.83e4·15-s + 1.09e5·17-s + 4.97e5i·19-s − 4.65e4i·21-s − 1.04e5·23-s + 1.72e6·25-s − 5.31e5i·27-s + 7.06e6i·29-s − 2.30e6·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.338i·5-s − 0.0905·7-s − 0.333·9-s − 1.49i·11-s + 0.144i·13-s + 0.195·15-s + 0.319·17-s + 0.875i·19-s − 0.0522i·21-s − 0.0780·23-s + 0.885·25-s − 0.192i·27-s + 1.85i·29-s − 0.448·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.782778153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.782778153\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
good | 5 | \( 1 + 473. iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 574.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.27e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.48e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.97e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 1.04e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.06e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 2.30e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.20e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 1.08e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.04e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 3.47e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.34e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 9.04e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 2.00e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.09e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.65e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 5.05e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.24e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.72e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.06e8T + 7.60e17T^{2} \) |
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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
−9.666119489714531236471145295099, −8.756092933277069396575757709207, −8.154585389704249016288044123102, −6.81244433751092106398524884663, −5.74628288899127445360026059299, −5.01170105915671368647078147615, −3.73073818384847725015953873396, −3.05666383082744309199179626362, −1.50891838598974429958691008274, −0.41667536918581748038528312373,
0.78464321964758404699970085552, 1.99251389552192638117603098122, 2.80930781000444961886047465498, 4.16545162140183838018348106344, 5.18925180660696784383106623452, 6.38756317689595828840468083374, 7.15376756025456075112219247016, 7.87644788940118557767895079468, 9.060988866793225241945901202027, 9.923388870652858383313233514075