| L(s) = 1 | − 12.2i·5-s − 14.2·7-s + 104. i·11-s + 75.2·13-s − 341. i·17-s − 706.·19-s + 596. i·23-s + 474.·25-s + 1.30e3i·29-s + 1.02e3·31-s + 175. i·35-s + 563.·37-s − 99.1i·41-s − 896.·43-s + 430. i·47-s + ⋯ |
| L(s) = 1 | − 0.491i·5-s − 0.291·7-s + 0.860i·11-s + 0.445·13-s − 1.18i·17-s − 1.95·19-s + 1.12i·23-s + 0.758·25-s + 1.54i·29-s + 1.07·31-s + 0.143i·35-s + 0.411·37-s − 0.0589i·41-s − 0.484·43-s + 0.194i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.187297986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187297986\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 12.2iT - 625T^{2} \) |
| 7 | \( 1 + 14.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 104. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 75.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 341. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 706.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 596. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.02e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 563.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 99.1iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 896.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 430. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.27e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.63e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.35e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.13e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.50e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.72e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.44e3T + 8.85e7T^{2} \) |
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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
−11.19269546101689711672096513950, −10.25443970525503768685104541800, −9.273310850138006740695461205740, −8.529527294232990013682017898718, −7.31651564992700556228294716008, −6.42445509774518911297906709428, −5.13015712684861997547887326623, −4.21410915283703071753401570865, −2.74428653612920223552533495702, −1.27035578701988277326846755452,
0.37270013246158453099451678998, 2.16315295724526497017142371644, 3.44141129426084342059414891211, 4.54219553276456518447167748428, 6.26991975764851449957783017003, 6.43097594108935883165572247086, 8.163614044981322327043538147989, 8.610371966802808560737501193715, 10.03950518675950028904928371796, 10.71140786934388147889038282121
