L(s) = 1 | + (−3.03 − 4.77i)2-s + 23.6·3-s + (−13.5 + 28.9i)4-s + (−71.7 − 112. i)6-s + 160. i·7-s + (179. − 23.4i)8-s + 314.·9-s + 129. i·11-s + (−319. + 684. i)12-s − 759.·13-s + (766. − 488. i)14-s + (−657. − 785. i)16-s + 323. i·17-s + (−955. − 1.50e3i)18-s + 198. i·19-s + ⋯ |
L(s) = 1 | + (−0.537 − 0.843i)2-s + 1.51·3-s + (−0.423 + 0.906i)4-s + (−0.813 − 1.27i)6-s + 1.23i·7-s + (0.991 − 0.129i)8-s + 1.29·9-s + 0.321i·11-s + (−0.641 + 1.37i)12-s − 1.24·13-s + (1.04 − 0.665i)14-s + (−0.641 − 0.766i)16-s + 0.271i·17-s + (−0.694 − 1.09i)18-s + 0.126i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.682769081\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.682769081\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.03 + 4.77i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 23.6T + 243T^{2} \) |
| 7 | \( 1 - 160. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 129. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 759.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 323. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 198. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.98e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.69e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.87e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.35e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.85e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.12e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e4iT - 8.58e9T^{2} \) |
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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
−11.98915583810508172301929370163, −10.56011895639764673632477185089, −9.451268735494979771698106721533, −9.032243148245899388907727299302, −8.096440955524923164120955604086, −7.19476083065170710525715748985, −5.09964071300648449967515285556, −3.60249555210829548988448042078, −2.59170669656295904191337193501, −1.81470995118566292108454931837,
0.47538981989869601644240840309, 2.12891173276333254892441113112, 3.71547782641945028220968528192, 4.89658188611473972524300267264, 6.67277156341076395654848102632, 7.58356667305928671164170155184, 8.148440166398344449259919438094, 9.344097590507781296176306100976, 9.903375788356589369260906155588, 10.99047157637860959826695658048