| L(s) = 1 | + 6.36·2-s + 15.5·3-s − 23.5·4-s − 60.2i·5-s + 99.1·6-s − 233. i·7-s − 556.·8-s + 243·9-s − 383. i·10-s − 1.39e3i·11-s − 366.·12-s − 1.28e3·13-s − 1.48e3i·14-s − 938. i·15-s − 2.03e3·16-s − 2.82e3i·17-s + ⋯ |
| L(s) = 1 | + 0.795·2-s + 0.577·3-s − 0.367·4-s − 0.481i·5-s + 0.459·6-s − 0.681i·7-s − 1.08·8-s + 0.333·9-s − 0.383i·10-s − 1.04i·11-s − 0.212·12-s − 0.584·13-s − 0.541i·14-s − 0.278i·15-s − 0.497·16-s − 0.574i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.22319 - 1.71797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22319 - 1.71797i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (3.98e3 - 1.14e4i)T \) |
| good | 2 | \( 1 - 6.36T + 64T^{2} \) |
| 5 | \( 1 + 60.2iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 233. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.39e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.28e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.82e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.01e4iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 5.52e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.73e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.97e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.84e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 3.09e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 6.27e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 5.61e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.59e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.44e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.37e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 4.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 7.20e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.85e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.74e5iT - 8.32e11T^{2} \) |
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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
−13.47968252188747266491869235016, −12.44642635557483225413400528749, −11.10619084294039885046358964641, −9.500606298760684485975658190484, −8.658765651648502833280870787735, −7.17623064990069687551016119572, −5.44218729573273615762669763112, −4.27268659804149311518893924540, −2.98002116783254031094925420087, −0.59807413347618245643268688086,
2.26426988140423536387959307621, 3.69584023690210714439087432445, 5.02773610936421637127824032589, 6.47578006349228613554866600895, 8.049061803659949061030031927531, 9.273929985450382536845131710468, 10.34861457508160992845445446344, 12.23125661490161814217370150207, 12.63650447333895532751056681175, 14.07706658454474377474145770641
