L(s) = 1 | + (14.2 + 8.20i)5-s + (23.8 + 41.2i)7-s + (95.2 − 54.9i)11-s + (17.3 − 30.0i)13-s + 208. i·17-s + 272.·19-s + (74.2 + 42.8i)23-s + (−177. − 308. i)25-s + (−552. + 318. i)29-s + (−549. + 951. i)31-s + 782. i·35-s + 219.·37-s + (2.86e3 + 1.65e3i)41-s + (−1.10e3 − 1.91e3i)43-s + (1.57e3 − 911. i)47-s + ⋯ |
L(s) = 1 | + (0.568 + 0.328i)5-s + (0.486 + 0.842i)7-s + (0.787 − 0.454i)11-s + (0.102 − 0.177i)13-s + 0.720i·17-s + 0.754·19-s + (0.140 + 0.0810i)23-s + (−0.284 − 0.493i)25-s + (−0.656 + 0.379i)29-s + (−0.571 + 0.990i)31-s + 0.638i·35-s + 0.159·37-s + (1.70 + 0.984i)41-s + (−0.599 − 1.03i)43-s + (0.715 − 0.412i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.440961770\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.440961770\) |
\(L(3)\) |
|
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 | \( 1 + (-14.2 - 8.20i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-23.8 - 41.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-95.2 + 54.9i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-17.3 + 30.0i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 208. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 272.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-74.2 - 42.8i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (552. - 318. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (549. - 951. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 219.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.86e3 - 1.65e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.10e3 + 1.91e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.57e3 + 911. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 4.21e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-3.96e3 - 2.28e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (227. + 394. i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.48e3 - 6.04e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.61e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.84e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (2.36e3 + 4.10e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (3.90e3 - 2.25e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 5.24e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (4.52e3 + 7.83e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−11.16098147645852531315756258176, −10.20583915115801974067794491933, −9.159396638741291122742823281798, −8.470307642754955336101559793060, −7.23633421910125936091345913999, −6.06859716554324315294864149319, −5.39542842398358729205602461247, −3.89041187326583531545168310545, −2.55776556415717188096934029414, −1.30786549103588220869015308933,
0.824973922732602697856840719784, 2.00113117197493232203982518786, 3.72731113420215117841767736504, 4.76332689592860738129507811082, 5.86309577964564749463182263914, 7.08530295555200668338571556130, 7.83318682882888188212852812009, 9.277284067949435830600357548886, 9.651060179073882590233685203203, 10.94908031856422234343780233349