L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 29.1i·5-s + 22.6i·8-s + 82.3·10-s + 1.41i·11-s − 82.3·13-s + 64.0·16-s + 262. i·17-s − 453.·19-s − 232. i·20-s + 4.00·22-s − 63.6i·23-s − 223·25-s + 232. i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.16i·5-s + 0.353i·8-s + 0.823·10-s + 0.0116i·11-s − 0.487·13-s + 0.250·16-s + 0.906i·17-s − 1.25·19-s − 0.582i·20-s + 0.00826·22-s − 0.120i·23-s − 0.356·25-s + 0.344i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3801988128\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3801988128\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 29.1iT - 625T^{2} \) |
| 11 | \( 1 - 1.41iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 82.3T + 2.85e4T^{2} \) |
| 17 | \( 1 - 262. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 453.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 63.6iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 502. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.27e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.56e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.31e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 328T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.13e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.01e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 757. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.19e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.13e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 422. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.65e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.97e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.33e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−9.408701064095503780969015993968, −8.493697373764983604532946308554, −7.53343282619650232468948770161, −6.64339525716463941257827413237, −5.78118175206049894774872281457, −4.51599013269403022076640406116, −3.59533640703128959160894617004, −2.64399587532939905400777985427, −1.74497786122304326811342273651, −0.096675206597797361651727388960,
0.946339761930522299084069044958, 2.36409488237303216540581207121, 3.93344531914360736715766275115, 4.76472783879323332505856011094, 5.46409291015710883613888634335, 6.46145331616278844822113290058, 7.43538765562739742975382923581, 8.211205430405846825607957080173, 9.026463682668123027403792628176, 9.545606870568718177847645410225