| L(s) = 1 | − 5.19·3-s + 8.72·5-s − 1.45i·7-s + 27·9-s + (62.2 − 103. i)11-s − 162. i·13-s − 45.3·15-s − 189. i·17-s + 590. i·19-s + 7.57i·21-s + 12.8·23-s − 548.·25-s − 140.·27-s + 282. i·29-s + 304.·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.349·5-s − 0.0297i·7-s + 0.333·9-s + (0.514 − 0.857i)11-s − 0.959i·13-s − 0.201·15-s − 0.656i·17-s + 1.63i·19-s + 0.0171i·21-s + 0.0243·23-s − 0.878·25-s − 0.192·27-s + 0.335i·29-s + 0.316·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.108105982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108105982\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19T \) |
| 11 | \( 1 + (-62.2 + 103. i)T \) |
| good | 5 | \( 1 - 8.72T + 625T^{2} \) |
| 7 | \( 1 + 1.45iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 162. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 189. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 590. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 12.8T + 2.79e5T^{2} \) |
| 29 | \( 1 - 282. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 304.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 464.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.19e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.59e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.82e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.02e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.48e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 356. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 8.25e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.97e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 5.78e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.13e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.33e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.12e4T + 8.85e7T^{2} \) |
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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
−10.13286325485690629030965171652, −9.135671398818409401384387788056, −8.127223048296374425065446674298, −7.20228284039065925646772445871, −5.93392726355238866400777811644, −5.62086760806161110595024312539, −4.20008214466777462198207653674, −3.09663630008384763166420782819, −1.54571429052643480669467400780, −0.32297364978766454038814305408,
1.28439546739796148458096700079, 2.42704077265723914757553172484, 4.10011072195342540785718322307, 4.83250048505010042944644978694, 6.08663609613832043241143199472, 6.75098139040458004122586715393, 7.70333239957731306877453189591, 9.045201406285479188360349551691, 9.597784257322400707696289625526, 10.58009285791333951525190628731
