L(s) = 1 | + (0.5 + 0.866i)2-s + (2.5 + 4.33i)3-s + (3.5 − 6.06i)4-s + (−2.5 − 4.33i)5-s + (−2.5 + 4.33i)6-s + 22·7-s + 15·8-s + (0.999 − 1.73i)9-s + (2.5 − 4.33i)10-s + 9·11-s + 35·12-s + (−27 + 46.7i)13-s + (11 + 19.0i)14-s + (12.5 − 21.6i)15-s + (−20.5 − 35.5i)16-s + (27 + 46.7i)17-s + ⋯ |
L(s) = 1 | + (0.176 + 0.306i)2-s + (0.481 + 0.833i)3-s + (0.437 − 0.757i)4-s + (−0.223 − 0.387i)5-s + (−0.170 + 0.294i)6-s + 1.18·7-s + 0.662·8-s + (0.0370 − 0.0641i)9-s + (0.0790 − 0.136i)10-s + 0.246·11-s + 0.841·12-s + (−0.576 + 0.997i)13-s + (0.209 + 0.363i)14-s + (0.215 − 0.372i)15-s + (−0.320 − 0.554i)16-s + (0.385 + 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.25507 + 0.487383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25507 + 0.487383i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.5 + 4.33i)T \) |
| 19 | \( 1 + (66.5 - 49.3i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \) |
| 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 - 9T + 1.33e3T^{2} \) |
| 13 | \( 1 + (27 - 46.7i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-27 - 46.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 23 | \( 1 + (-46 + 79.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-67 + 116. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 252T + 2.97e4T^{2} \) |
| 37 | \( 1 + 236T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-121.5 - 210. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (248 + 429. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (251 - 434. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (31 - 53.6i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (340.5 + 589. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-71 + 122. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (27.5 - 47.6i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-487 - 843. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (347.5 + 601. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-368 - 637. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 63T + 5.71e5T^{2} \) |
| 89 | \( 1 + (363 - 628. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-583.5 - 1.01e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.21712530343059068595670031907, −12.44058131781760656095872224115, −11.27649951212410368447791092906, −10.31272078955377143794714183235, −9.218415360922711377934697209765, −8.079984804409519744365686719613, −6.62523077144777768688396095426, −5.05702198807915921117737565279, −4.13421292185354867165874738185, −1.74699357876445148617334030237,
1.81864937723505996313741253251, 3.13095906600118597961816098731, 4.92045739555757377399174611867, 7.06685413320404984814767140528, 7.65583369751070102026145787483, 8.604153506381280619778995602243, 10.56107768258407645578290151821, 11.44486107416496694412240750897, 12.42966042577659884862406238900, 13.27845319861924151200689279822