| L(s) = 1 | + (7.74 − 2i)2-s + 30.9·3-s + (56.0 − 30.9i)4-s + (240. − 61.9i)6-s + 309.·7-s + (371. − 352. i)8-s + 231.·9-s − 960. i·11-s + (1.73e3 − 960. i)12-s + 1.46e3i·13-s + (2.40e3 − 619. i)14-s + (2.17e3 − 3.47e3i)16-s + 4.76e3i·17-s + (1.78e3 − 462. i)18-s − 7.52e3i·19-s + ⋯ |
| L(s) = 1 | + (0.968 − 0.250i)2-s + 1.14·3-s + (0.875 − 0.484i)4-s + (1.11 − 0.286i)6-s + 0.903·7-s + (0.726 − 0.687i)8-s + 0.316·9-s − 0.721i·11-s + (1.00 − 0.555i)12-s + 0.667i·13-s + (0.874 − 0.225i)14-s + (0.531 − 0.847i)16-s + 0.970i·17-s + (0.306 − 0.0792i)18-s − 1.09i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(5.31603 - 1.64965i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.31603 - 1.64965i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.74 + 2i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 30.9T + 729T^{2} \) |
| 7 | \( 1 - 309.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 960. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.46e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 4.76e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.52e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.04e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.54e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.18e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.99e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 2.93e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 2.15e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.56e3T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.92e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 7.84e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.09e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.94e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.32e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.88e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.10e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.04e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.10e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.45e6iT - 8.32e11T^{2} \) |
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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
−12.92230421715513855866327041604, −11.52548208136601991637057336568, −10.80161682126046102257404893036, −9.191279738666310674007792472977, −8.187494664840121359859080205494, −6.89907817215498635287404978846, −5.35235398090741338093107819591, −4.01617129627143305954723124904, −2.81245791639367192978583079829, −1.54787878125178688792117040420,
1.88070172521458756778420041853, 3.07042784105812711193498454428, 4.40050163381867256544109044356, 5.66146650791360751532750261411, 7.43503405062434192912063162683, 8.004912031920310356615661407378, 9.377616828234887650708775037418, 10.88829878633979917815087471350, 11.97503871732445504130142301494, 13.09749858943544587965853158779
