L(s) = 1 | + 12i·3-s + 54i·5-s + 88·7-s + 99·9-s + 540i·11-s + 418i·13-s − 648·15-s + 594·17-s − 836i·19-s + 1.05e3i·21-s + 4.10e3·23-s + 209·25-s + 4.10e3i·27-s + 594i·29-s + 4.25e3·31-s + ⋯ |
L(s) = 1 | + 0.769i·3-s + 0.965i·5-s + 0.678·7-s + 0.407·9-s + 1.34i·11-s + 0.685i·13-s − 0.743·15-s + 0.498·17-s − 0.531i·19-s + 0.522i·21-s + 1.61·23-s + 0.0668·25-s + 1.08i·27-s + 0.131i·29-s + 0.795·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.341613047\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.341613047\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 12iT - 243T^{2} \) |
| 5 | \( 1 - 54iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 88T + 1.68e4T^{2} \) |
| 11 | \( 1 - 540iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 418iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 594T + 1.41e6T^{2} \) |
| 19 | \( 1 + 836iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 594iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 298iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.94e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.66e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.47e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.18e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.77e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e5T + 8.58e9T^{2} \) |
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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
−11.30640195795238470713041236165, −10.49512921935053364460954883036, −9.800847885907381303845183186686, −8.801830771852822067513318981193, −7.31579272115181977647119822190, −6.81825283350181482479339067061, −5.07680719835353065028031038376, −4.34721650962262878894440722553, −2.99849158018346843442178430617, −1.58855452875001172237472826631,
0.73608196413052327172784557268, 1.42841803377364697087145988828, 3.16123620635294066555346658567, 4.71642954700158013437737952717, 5.63731839674385602910431264745, 6.85982653732257161847051811756, 8.116790953614393569051454657527, 8.463440606133406618673717390408, 9.800052300470994582882185240025, 10.96673501532699991965900545132