L(s) = 1 | + 1.41i·2-s + (−0.821 + 2.88i)3-s − 2.00·4-s + (−4.08 − 1.16i)6-s + 2.64·7-s − 2.82i·8-s + (−7.64 − 4.74i)9-s + 18.1i·11-s + (1.64 − 5.77i)12-s + 22.5·13-s + 3.74i·14-s + 4.00·16-s − 0.457i·17-s + (6.70 − 10.8i)18-s + 30.1·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.273 + 0.961i)3-s − 0.500·4-s + (−0.680 − 0.193i)6-s + 0.377·7-s − 0.353i·8-s + (−0.849 − 0.526i)9-s + 1.65i·11-s + (0.136 − 0.480i)12-s + 1.73·13-s + 0.267i·14-s + 0.250·16-s − 0.0268i·17-s + (0.372 − 0.600i)18-s + 1.58·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.763209639\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763209639\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (0.821 - 2.88i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 - 18.1iT - 121T^{2} \) |
| 13 | \( 1 - 22.5T + 169T^{2} \) |
| 17 | \( 1 + 0.457iT - 289T^{2} \) |
| 19 | \( 1 - 30.1T + 361T^{2} \) |
| 23 | \( 1 - 12.3iT - 529T^{2} \) |
| 29 | \( 1 - 4.70iT - 841T^{2} \) |
| 31 | \( 1 - 45.2T + 961T^{2} \) |
| 37 | \( 1 + 32.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 22.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.67iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 20.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 97.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 121. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 151.T + 9.40e3T^{2} \) |
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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
−9.958921093666518094761859379000, −9.299905804370612214352981506614, −8.491508569186945994210750992001, −7.60646022989965382765964239361, −6.65759028780548529085508844425, −5.73285482569279568640077420327, −4.93672429659346924066617900390, −4.18088295347253028729682283897, −3.18109563631594288631688801289, −1.30812097605279049331012281590,
0.68723362475501690343582978294, 1.42590548909866247447317292652, 2.88176602052346747786818084582, 3.65532070371441034488671764523, 5.14300158844107857628155660777, 5.91416993437269436301634464800, 6.66662825381452027276394600716, 8.039088070209803533868692294730, 8.355894699366726234232258446937, 9.219545835697895836516837119749