L(s) = 1 | + (4.41 − 26.6i)3-s − 127.·5-s + 467.·7-s + (−689. − 235. i)9-s − 554.·11-s − 357. i·13-s + (−561. + 3.38e3i)15-s − 4.49e3i·17-s − 3.24e3i·19-s + (2.06e3 − 1.24e4i)21-s + 1.74e4i·23-s + 511.·25-s + (−9.31e3 + 1.73e4i)27-s − 1.74e4·29-s + 8.03e3·31-s + ⋯ |
L(s) = 1 | + (0.163 − 0.986i)3-s − 1.01·5-s + 1.36·7-s + (−0.946 − 0.322i)9-s − 0.416·11-s − 0.162i·13-s + (−0.166 + 1.00i)15-s − 0.915i·17-s − 0.472i·19-s + (0.223 − 1.34i)21-s + 1.43i·23-s + 0.0327·25-s + (−0.473 + 0.880i)27-s − 0.714·29-s + 0.269·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4680344536\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4680344536\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.41 + 26.6i)T \) |
good | 5 | \( 1 + 127.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 467.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 554.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 357. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.49e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.24e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.74e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.74e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.03e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.94e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.18e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.54e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.90e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.06e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.51e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.02e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.62e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.49e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.18e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.38e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.22e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.15e6T + 8.32e11T^{2} \) |
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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
−11.01497064000909990665749883016, −9.397765787990849710599835717184, −8.363245433678442498083760772256, −7.65516534169441402917656678337, −7.23530297779929627687524190274, −5.70778513200794818018619402561, −4.75408405768505272494027713213, −3.44213424905554442978442141685, −2.18419831640837436865958123188, −1.04272332817705470681669316290,
0.11224284858795867195704544089, 1.83119211808523495219119848869, 3.26831870144606819490372044645, 4.33768968591169505772729676521, 4.87170678126632180800292745683, 6.15968673688626690826032528674, 7.84786608935188167140144824387, 8.116921302248323433029332905961, 9.093330352814439093110535021924, 10.39121193339745174710659243819