L(s) = 1 | + 81i·3-s + 2.17e3i·5-s + 7.65e3·7-s − 6.56e3·9-s + 6.14e4i·11-s − 2.82e4i·13-s − 1.76e5·15-s + 1.93e5·17-s − 3.66e5i·19-s + 6.20e5i·21-s + 8.20e5·23-s − 2.79e6·25-s − 5.31e5i·27-s + 4.44e6i·29-s − 2.59e6·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.55i·5-s + 1.20·7-s − 0.333·9-s + 1.26i·11-s − 0.274i·13-s − 0.899·15-s + 0.560·17-s − 0.645i·19-s + 0.696i·21-s + 0.611·23-s − 1.42·25-s − 0.192i·27-s + 1.16i·29-s − 0.505·31-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
\(L(5)\) |
\(\approx\) |
\(1.335925335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.335925335\) |
\(L(\frac{11}{2})\) |
|
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 | \( 1 - 2.17e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 7.65e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.14e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 2.82e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.93e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.66e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 8.20e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.44e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 2.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.14e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 3.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 9.42e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 3.59e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.97e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 9.36e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 3.86e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.80e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 1.54e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.79e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.21e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.93e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 1.05e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.91e8T + 7.60e17T^{2} \) |
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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
−10.48930310751275347356776797059, −9.705417469314365284889336556086, −8.554067665327048844667421876145, −7.39816544067826594944232008056, −6.93894009129631104847533738707, −5.51569284624626132417759884024, −4.67637933616450734835122017920, −3.53908614957340362483994656898, −2.56520585105877906452247326422, −1.55738163827896759980346924457,
0.23117516619508811698304751487, 1.20015723519305237350328197586, 1.73037298776615854183797815340, 3.34849318874416848711015088993, 4.65781948150422377207137991386, 5.31434059915903920999918902151, 6.26778124807129512347289634439, 7.75578174491332994984837729711, 8.329676196219061271710301387613, 8.875073793615873183237238248565