L(s) = 1 | + 2i·5-s + 7-s + 5i·13-s + 17-s − 4i·19-s + 5·23-s + 25-s − 9i·29-s + 7·31-s + 2i·35-s + 2i·37-s + 2·41-s − 5i·43-s + 49-s − 9i·53-s + ⋯ |
L(s) = 1 | + 0.894i·5-s + 0.377·7-s + 1.38i·13-s + 0.242·17-s − 0.917i·19-s + 1.04·23-s + 0.200·25-s − 1.67i·29-s + 1.25·31-s + 0.338i·35-s + 0.328i·37-s + 0.312·41-s − 0.762i·43-s + 0.142·49-s − 1.23i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.243389270\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.243389270\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 13T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.160121445874212267566844211224, −7.29296081509303595524054842237, −6.75885488534285347256875211226, −6.27963530035108298059024491788, −5.21006649429812336758834395192, −4.52744584124502477203492833870, −3.76819757261829282929161123519, −2.75163516912380957984230910762, −2.16884720685062768093969796126, −0.890840586339115948092554026183,
0.78187672482828431219914192437, 1.46644659443831966827454615355, 2.77789726791239328209640723026, 3.46982435331385974920950612120, 4.53267727581114999926143974025, 5.12834744773377447375616099912, 5.64927597714867348917153816715, 6.55062231685487687322961879923, 7.43015231496483898726799062342, 8.129666030689296445274998829153