L(s) = 1 | − 1.50·2-s + (2.08 − 2.15i)3-s − 1.72·4-s + (4.79 + 1.41i)5-s + (−3.14 + 3.24i)6-s − 2.64i·7-s + 8.63·8-s + (−0.288 − 8.99i)9-s + (−7.22 − 2.13i)10-s + 0.491i·11-s + (−3.60 + 3.72i)12-s − 21.9i·13-s + 3.98i·14-s + (13.0 − 7.37i)15-s − 6.11·16-s − 4.43·17-s + ⋯ |
L(s) = 1 | − 0.753·2-s + (0.695 − 0.718i)3-s − 0.431·4-s + (0.958 + 0.283i)5-s + (−0.524 + 0.541i)6-s − 0.377i·7-s + 1.07·8-s + (−0.0321 − 0.999i)9-s + (−0.722 − 0.213i)10-s + 0.0446i·11-s + (−0.300 + 0.310i)12-s − 1.68i·13-s + 0.284i·14-s + (0.870 − 0.491i)15-s − 0.381·16-s − 0.260·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.999546 - 0.583552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999546 - 0.583552i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.08 + 2.15i)T \) |
| 5 | \( 1 + (-4.79 - 1.41i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 2 | \( 1 + 1.50T + 4T^{2} \) |
| 11 | \( 1 - 0.491iT - 121T^{2} \) |
| 13 | \( 1 + 21.9iT - 169T^{2} \) |
| 17 | \( 1 + 4.43T + 289T^{2} \) |
| 19 | \( 1 - 14.4T + 361T^{2} \) |
| 23 | \( 1 - 19.9T + 529T^{2} \) |
| 29 | \( 1 - 35.6iT - 841T^{2} \) |
| 31 | \( 1 + 45.5T + 961T^{2} \) |
| 37 | \( 1 - 58.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 45.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.94iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 9.08iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 37.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 4.57iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 14.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 155. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124. iT - 9.40e3T^{2} \) |
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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
−13.30266291144965519706074946810, −12.77460031751706092254631749326, −10.85599109284849560425315129556, −9.897932811585331011006868028168, −9.012488454562743651290701007572, −7.942936644115400915213391232244, −6.94704233708344429546699981597, −5.32233942477268280394531614153, −3.13515086441010493483666631257, −1.22154335826275827376331484453,
2.03043743402729768536883375576, 4.18002784852237434316685526309, 5.42889178185777499500198965329, 7.32484939827684310011387510877, 8.943397140323679980665271008334, 9.098780433283967606118377377045, 10.06644662722712975292734795310, 11.23252440851386349034515192347, 12.94879812392266398361875077602, 13.90061398862607556247598181476