| L(s) = 1 | − 22.6i·2-s + (200. + 137. i)3-s − 512.·4-s + (3.11e3 − 4.53e3i)6-s + 2.32e4·7-s + 1.15e4i·8-s + (2.11e4 + 5.51e4i)9-s + 6.24e4i·11-s + (−1.02e5 − 7.04e4i)12-s + 1.70e5·13-s − 5.25e5i·14-s + 2.62e5·16-s + 2.66e6i·17-s + (1.24e6 − 4.78e5i)18-s + 7.66e5·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.824 + 0.566i)3-s − 0.500·4-s + (0.400 − 0.582i)6-s + 1.38·7-s + 0.353i·8-s + (0.358 + 0.933i)9-s + 0.387i·11-s + (−0.412 − 0.283i)12-s + 0.458·13-s − 0.977i·14-s + 0.250·16-s + 1.87i·17-s + (0.660 − 0.253i)18-s + 0.309·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.228642045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.228642045\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 22.6iT \) |
| 3 | \( 1 + (-200. - 137. i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.32e4T + 2.82e8T^{2} \) |
| 11 | \( 1 - 6.24e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.70e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.66e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 7.66e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + 1.40e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 4.83e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 4.18e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 5.01e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.49e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.98e8T + 2.16e16T^{2} \) |
| 47 | \( 1 - 1.55e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 4.21e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 2.92e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 5.30e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 5.22e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 5.71e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.18e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 1.96e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 2.18e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 2.38e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 8.84e9T + 7.37e19T^{2} \) |
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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
−10.88878941339414347358490981877, −10.55302829518912159295592222983, −9.169760665949521276191634270414, −8.422207647731094901967849573559, −7.53850685072409387554957524605, −5.55197771532228809131311499033, −4.41309875019210982809589957407, −3.61586297913753360211912752738, −2.12405714639516977772720686862, −1.42292055874586782159086811533,
0.62892339417492159160111988290, 1.74369722639636817065349974674, 3.15102041899732652124873586147, 4.53314115514720800505450552414, 5.66279532712221641200800959758, 7.09348281082224404867007563707, 7.75829104749286446338275372751, 8.681093928658695972370861196721, 9.493226477050278940361958253764, 11.09285238509102828178674215392
