| L(s) = 1 | + 51.1·3-s − 142. i·5-s + 217. i·7-s + 1.89e3·9-s − 25.7·11-s − 1.39e3i·13-s − 7.30e3i·15-s + 4.00e3·17-s − 129.·19-s + 1.11e4i·21-s − 3.69e3i·23-s − 4.75e3·25-s + 5.95e4·27-s + 4.08e4i·29-s − 4.61e4i·31-s + ⋯ |
| L(s) = 1 | + 1.89·3-s − 1.14i·5-s + 0.634i·7-s + 2.59·9-s − 0.0193·11-s − 0.632i·13-s − 2.16i·15-s + 0.814·17-s − 0.0188·19-s + 1.20i·21-s − 0.303i·23-s − 0.304·25-s + 3.02·27-s + 1.67i·29-s − 1.54i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.701062642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.701062642\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 51.1T + 729T^{2} \) |
| 5 | \( 1 + 142. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 217. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 25.7T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.39e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.00e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 129.T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.69e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.08e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 6.59e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.18e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 3.00e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.97e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.37e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.00e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.43e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.06e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.49e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.66e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.52e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.12e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.97e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 8.78e3T + 8.32e11T^{2} \) |
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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
−10.53063895242180312985831179442, −9.400168487341648504820316960734, −8.928636712332356440518384333677, −8.139131114503097551912728951054, −7.35803536112275390991379017749, −5.59330388505684489026190098384, −4.38424293751647697918565588177, −3.26171066802670981283651099175, −2.19208899178517823929466567952, −1.01132917359906854116457908149,
1.42679877506258174201580861230, 2.66868973021685771431835726803, 3.41164469763660828301806103448, 4.40924151151587955730634638419, 6.50112816979013301337246998974, 7.38687941154751441608152317490, 8.026577255000933807437431453451, 9.188280588568109953567260021533, 9.988720665499127036194650614735, 10.74368210873826673448095364711
