| L(s) = 1 | − 13.1i·3-s − 61.6·7-s − 91.5·9-s − 182.·11-s + 141.·13-s − 40.3i·17-s − 321.·19-s + 809. i·21-s − 816.·23-s + 138. i·27-s − 1.19e3i·29-s + 361. i·31-s + 2.39e3i·33-s − 0.0209·37-s − 1.86e3i·39-s + ⋯ |
| L(s) = 1 | − 1.45i·3-s − 1.25·7-s − 1.12·9-s − 1.50·11-s + 0.838·13-s − 0.139i·17-s − 0.890·19-s + 1.83i·21-s − 1.54·23-s + 0.189i·27-s − 1.42i·29-s + 0.376i·31-s + 2.19i·33-s − 1.52e − 5·37-s − 1.22i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5968261215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5968261215\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 13.1iT - 81T^{2} \) |
| 7 | \( 1 + 61.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 182.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 141.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 40.3iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 321.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 816.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.19e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 361. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 0.0209T + 1.87e6T^{2} \) |
| 41 | \( 1 + 872.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 3.12e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.65e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.95e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.41e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.25e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.35e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 660. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.94e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.62e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.52e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.94e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 679. iT - 8.85e7T^{2} \) |
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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
−9.799589447081457836973796677823, −8.501917300122027481791811640678, −7.957126268849100945609803418268, −7.06652645074044052589272144522, −6.23092200932113509965026582400, −5.73259152544991928297435781411, −4.11467491492820987468543540673, −2.84118608705108982408317028783, −2.07174179901394854900230796956, −0.64097840774818177837666306991,
0.20738417081511124396963897595, 2.35178131783266834034525928849, 3.46979225077749585688355338518, 4.05107687208905419134367995484, 5.25584481077056159030884270627, 5.92846509055166384871821868962, 7.02269350124486191402654546240, 8.300609489188615820037045582374, 8.951572349482876397431820680518, 9.933882436174443296075240362601
