L(s) = 1 | + (12.4 − 12.4i)3-s + (55 − 10i)5-s + (136. + 136. i)7-s − 67i·9-s + 124. i·11-s + (165 + 165i)13-s + (560. − 809. i)15-s + (1.26e3 − 1.26e3i)17-s − 2.73e3·19-s + 3.40e3·21-s + (−535. + 535. i)23-s + (2.92e3 − 1.10e3i)25-s + (2.19e3 + 2.19e3i)27-s − 2.59e3i·29-s − 7.09e3i·31-s + ⋯ |
L(s) = 1 | + (0.798 − 0.798i)3-s + (0.983 − 0.178i)5-s + (1.05 + 1.05i)7-s − 0.275i·9-s + 0.310i·11-s + (0.270 + 0.270i)13-s + (0.642 − 0.928i)15-s + (1.06 − 1.06i)17-s − 1.74·19-s + 1.68·21-s + (−0.211 + 0.211i)23-s + (0.936 − 0.352i)25-s + (0.578 + 0.578i)27-s − 0.573i·29-s − 1.32i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.91515 - 0.551157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.91515 - 0.551157i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-55 + 10i)T \) |
good | 3 | \( 1 + (-12.4 + 12.4i)T - 243iT^{2} \) |
| 7 | \( 1 + (-136. - 136. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 124. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-165 - 165i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.26e3 + 1.26e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (535. - 535. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 2.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.09e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.38e3 + 1.38e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 2.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.38e4 - 1.38e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (4.94e3 + 4.94e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.39e4 + 2.39e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 2.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.68e4 - 2.68e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 6.71e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.16e4 + 2.16e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.87e4 + 3.87e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.14e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (615 - 615i)T - 8.58e9iT^{2} \) |
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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
−13.40603671591396989314076243712, −12.49547295252474549843368481608, −11.31563187857646829452604383847, −9.733459718758168601843610796225, −8.656130079463445008393658399599, −7.80782899659607883604493348634, −6.23844461851766119440134984374, −4.93057120335512034316683806181, −2.49332463228789660054358357498, −1.65690004204520289834893031218,
1.56306156883154118399171942103, 3.41030924230557639299549446681, 4.70248523032514875324471775811, 6.32292792637191274986313408852, 8.018818912085021240917515774791, 8.947326297281858109067399224280, 10.41459626502379586880336013047, 10.63198519916267959311818968071, 12.59485088765987710831833300639, 13.89519227404656977871090907604