| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 1.41i·5-s + 2.82i·8-s − 2.00·10-s − 7.07i·11-s + 15·13-s + 4.00·16-s − 11.3i·17-s − 13·19-s + 2.82i·20-s − 10.0·22-s + 22.6i·23-s + 23·25-s − 21.2i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.282i·5-s + 0.353i·8-s − 0.200·10-s − 0.642i·11-s + 1.15·13-s + 0.250·16-s − 0.665i·17-s − 0.684·19-s + 0.141i·20-s − 0.454·22-s + 0.983i·23-s + 0.920·25-s − 0.815i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.426884238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426884238\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 25T^{2} \) |
| 11 | \( 1 + 7.07iT - 121T^{2} \) |
| 13 | \( 1 - 15T + 169T^{2} \) |
| 17 | \( 1 + 11.3iT - 289T^{2} \) |
| 19 | \( 1 + 13T + 361T^{2} \) |
| 23 | \( 1 - 22.6iT - 529T^{2} \) |
| 29 | \( 1 + 22.6iT - 841T^{2} \) |
| 31 | \( 1 - 3T + 961T^{2} \) |
| 37 | \( 1 - 17T + 1.36e3T^{2} \) |
| 41 | \( 1 + 80.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 85T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 33.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 90.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72T + 3.72e3T^{2} \) |
| 67 | \( 1 - 43T + 4.48e3T^{2} \) |
| 71 | \( 1 - 52.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 95T + 5.32e3T^{2} \) |
| 79 | \( 1 - 69T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 16T + 9.40e3T^{2} \) |
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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
−9.647689109089151303082928406159, −8.759400003704788231675182017771, −8.274869924013583988531479582796, −7.03183669353088332145770969657, −5.96909271287050820899040322624, −5.08831221549109123911256204597, −3.96546391951100975620875913794, −3.12239826401546163222934072125, −1.76748740565168009105784053164, −0.50210037649063395664577859525,
1.38706044404102170647020819007, 2.97975731402677073802145215034, 4.16355648055610461305929991480, 4.99037568207296011803907691133, 6.32571181011750943852912596866, 6.56689313010192626865148743191, 7.79454041246524013305767998474, 8.483446681638148964994144075885, 9.246404499572143332290508045652, 10.33483416973739514455699379715
