L(s) = 1 | − 20.6i·3-s + 25i·5-s − 93.5·7-s − 182.·9-s − 659. i·11-s − 183. i·13-s + 515.·15-s − 467.·17-s − 225. i·19-s + 1.92e3i·21-s + 1.10e3·23-s − 625·25-s − 1.24e3i·27-s + 1.73e3i·29-s − 7.27e3·31-s + ⋯ |
L(s) = 1 | − 1.32i·3-s + 0.447i·5-s − 0.721·7-s − 0.751·9-s − 1.64i·11-s − 0.301i·13-s + 0.591·15-s − 0.391·17-s − 0.143i·19-s + 0.954i·21-s + 0.437·23-s − 0.200·25-s − 0.329i·27-s + 0.383i·29-s − 1.35·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.1665308873\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1665308873\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 25iT \) |
good | 3 | \( 1 + 20.6iT - 243T^{2} \) |
| 7 | \( 1 + 93.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 659. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 183. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 467.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 225. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.73e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 4.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.23e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 619.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.07e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.06e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.19e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.26e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.55e4T + 8.58e9T^{2} \) |
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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
−10.28233148866741248073494772250, −8.966146979740868559029339218906, −8.137971796452777236281255363696, −7.05562992831971196871028876536, −6.43420337499020936331062890027, −5.49714548895891574668476670372, −3.56162377428977669839879273814, −2.62254517794785191316548404548, −1.16848387987432076999869294451, −0.04567840241889311487727436727,
1.96594322975377479839435599270, 3.53608099077541312947514437212, 4.43762503147368013867668950277, 5.21365416376646130525510778822, 6.58510079906017574217679376108, 7.67837491259437438679065729685, 9.246852725026220862157252830109, 9.411996337370035299446300228295, 10.35621867690437501557433905887, 11.23395574413496534574535425222