| L(s) = 1 | + 4i·2-s − 16·4-s + 79i·7-s − 64i·8-s − 150·11-s + 137i·13-s − 316·14-s + 256·16-s − 2.03e3i·17-s + 1.96e3·19-s − 600i·22-s + 1.35e3i·23-s − 548·26-s − 1.26e3i·28-s − 2.94e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.609i·7-s − 0.353i·8-s − 0.373·11-s + 0.224i·13-s − 0.430·14-s + 0.250·16-s − 1.70i·17-s + 1.25·19-s − 0.264i·22-s + 0.532i·23-s − 0.158·26-s − 0.304i·28-s − 0.650·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.822725586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.822725586\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 79iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 150T + 1.61e5T^{2} \) |
| 13 | \( 1 - 137iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.03e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.96e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.35e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 713T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.23e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.11e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.58e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.25e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.24e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.57e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.73e4iT - 8.58e9T^{2} \) |
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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
−10.19586273736828098567587011329, −9.382027357358779703889525254263, −8.656412518487574295970581946719, −7.51742450475416244530281258986, −6.92076699240343023262410813374, −5.54253124779550111429177309887, −5.10284250132423505466875919365, −3.63383486875965747768779267000, −2.40635113310196024430632740813, −0.72866644153454381022560630350,
0.67633483765294858848792265312, 1.82413190568939108822897292545, 3.17634843202695991795518528389, 4.08405332976486667182047685100, 5.20933996760708643589923353560, 6.30509212984696417430486857269, 7.58356466632206388908764870322, 8.324479894742143771783299054368, 9.452020188544148580476228338360, 10.28004838519339303610842711583
