L(s) = 1 | + 1.95·3-s + (−2.75 − 4.17i)5-s + 2.18i·7-s − 5.19·9-s − 9.50·11-s − 20.4·13-s + (−5.37 − 8.13i)15-s + 6.15i·17-s + (9.68 − 16.3i)19-s + 4.26i·21-s + 10.9i·23-s + (−9.82 + 22.9i)25-s − 27.6·27-s + 28.4i·29-s − 24.8i·31-s + ⋯ |
L(s) = 1 | + 0.650·3-s + (−0.550 − 0.834i)5-s + 0.312i·7-s − 0.577·9-s − 0.864·11-s − 1.57·13-s + (−0.358 − 0.542i)15-s + 0.362i·17-s + (0.509 − 0.860i)19-s + 0.203i·21-s + 0.475i·23-s + (−0.392 + 0.919i)25-s − 1.02·27-s + 0.980i·29-s − 0.800i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0485i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00653801 - 0.269414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00653801 - 0.269414i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.75 + 4.17i)T \) |
| 19 | \( 1 + (-9.68 + 16.3i)T \) |
good | 3 | \( 1 - 1.95T + 9T^{2} \) |
| 7 | \( 1 - 2.18iT - 49T^{2} \) |
| 11 | \( 1 + 9.50T + 121T^{2} \) |
| 13 | \( 1 + 20.4T + 169T^{2} \) |
| 17 | \( 1 - 6.15iT - 289T^{2} \) |
| 23 | \( 1 - 10.9iT - 529T^{2} \) |
| 29 | \( 1 - 28.4iT - 841T^{2} \) |
| 31 | \( 1 + 24.8iT - 961T^{2} \) |
| 37 | \( 1 + 37.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 81.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 46.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 94.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 102. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 73.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 91.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 84.3T + 9.40e3T^{2} \) |
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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
−10.69594760394910947764929106054, −9.479526709130210666902643192090, −8.875181169900738761868659633706, −7.922835230924070568698355840437, −7.26148849138152065231051008345, −5.52257173178637926465763212934, −4.83399359157059048612140914490, −3.37473713017714486696656205007, −2.23468901895107711676257622246, −0.10023122667104275332378500758,
2.45569087230764471771464694603, 3.21235741756006400021997413088, 4.57116734118940156450345083044, 5.84069694794426321841583026637, 7.26602415058479688454630905935, 7.69350480086851032165035237019, 8.705952969635995109467372235660, 9.943864815803041484875580510554, 10.49480615476979785893409229779, 11.66759528101252529876841609839