L(s) = 1 | + (−1.73 + i)2-s + (4.33 + 2.5i)3-s + (1.99 − 3.46i)4-s − 10·6-s + (12.1 + 14i)7-s + 7.99i·8-s + (−0.999 − 1.73i)9-s + (28.5 − 49.3i)11-s + (17.3 − 10i)12-s − 70i·13-s + (−35 − 12.1i)14-s + (−8 − 13.8i)16-s + (44.1 + 25.5i)17-s + (3.46 + 1.99i)18-s + (2.5 + 4.33i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.833 + 0.481i)3-s + (0.249 − 0.433i)4-s − 0.680·6-s + (0.654 + 0.755i)7-s + 0.353i·8-s + (−0.0370 − 0.0641i)9-s + (0.781 − 1.35i)11-s + (0.416 − 0.240i)12-s − 1.49i·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.630 + 0.363i)17-s + (0.0453 + 0.0261i)18-s + (0.0301 + 0.0522i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.043463599\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.043463599\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-12.1 - 14i)T \) |
good | 3 | \( 1 + (-4.33 - 2.5i)T + (13.5 + 23.3i)T^{2} \) |
| 11 | \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 70iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-44.1 - 25.5i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-59.7 + 34.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 114T + 2.43e4T^{2} \) |
| 31 | \( 1 + (11.5 - 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-219. + 126.5i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 42T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (174. - 100.5i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-340. - 196.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-109.5 + 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-354.5 - 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-362. - 209.5i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 96T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-271. - 156.5i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-230.5 - 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 588iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (508.5 + 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.83e3iT - 9.12e5T^{2} \) |
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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
−10.92720654449724531883873937273, −9.927466167581271268560896089683, −8.924811040116348888380544824960, −8.484682151683782212278700180534, −7.69112770945557749267908207364, −6.12152172169626307583141251825, −5.40313625862217504774070606473, −3.70967489476567341201369006987, −2.67944989146160715579611911805, −0.907444217699208045675598110339,
1.37239550059011243909243374687, 2.17042823440297574894102668864, 3.73538817850674558003899282502, 4.81588500129753911390184792299, 6.81887116582464786321185669613, 7.36304028606924211229960639457, 8.234301787699640926400302948393, 9.268127342853435264781346309105, 9.852382879242277658993549974134, 11.16755752930477829151232787084