| L(s) = 1 | − 9.72i·3-s + 25i·5-s − 65.1·7-s + 148.·9-s + 71.5i·11-s − 733. i·13-s + 243.·15-s − 716.·17-s + 2.33e3i·19-s + 633. i·21-s + 278.·23-s − 625·25-s − 3.80e3i·27-s − 2.95e3i·29-s + 6.04e3·31-s + ⋯ |
| L(s) = 1 | − 0.623i·3-s + 0.447i·5-s − 0.502·7-s + 0.610·9-s + 0.178i·11-s − 1.20i·13-s + 0.279·15-s − 0.600·17-s + 1.48i·19-s + 0.313i·21-s + 0.109·23-s − 0.200·25-s − 1.00i·27-s − 0.652i·29-s + 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.123371389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123371389\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
| good | 3 | \( 1 + 9.72iT - 243T^{2} \) |
| 7 | \( 1 + 65.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 71.5iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 733. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 716.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.33e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 278.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 230. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.24e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.28e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.94e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.39e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.59e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.02e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.20e5T + 8.58e9T^{2} \) |
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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
−10.28336732000480891313385109743, −9.786303618067550857628712949113, −8.308845415894962328308832025572, −7.54357213713412200956756077214, −6.59136235650274552311584040926, −5.73813619386020357273302859634, −4.21849711060272298870995726461, −3.00438125201692724597275028982, −1.72860699056619363049462756238, −0.30206689174511111334824196239,
1.31116634097681017148781427033, 2.88629840241886848215338095833, 4.27949425133814525158761109936, 4.85437147147949406555086265547, 6.38499349748774901965652513679, 7.15687721920142806019577197295, 8.637223664726668143076012188382, 9.297974084310863900995796108533, 10.08379295973419454532797890053, 11.12592146664757670262576327370
