| L(s) = 1 | − i·2-s + (−2 − 2.23i)3-s + 3·4-s + 4.38i·5-s + (−2.23 + 2i)6-s + 2.14·7-s − 7i·8-s + (−1.00 + 8.94i)9-s + 4.38·10-s + (−6 − 6.70i)12-s + 8.76·13-s − 2.14i·14-s + (9.79 − 8.76i)15-s + 5·16-s + 6i·17-s + (8.94 + 1.00i)18-s + ⋯ |
| L(s) = 1 | − 0.5i·2-s + (−0.666 − 0.745i)3-s + 0.750·4-s + 0.876i·5-s + (−0.372 + 0.333i)6-s + 0.306·7-s − 0.875i·8-s + (−0.111 + 0.993i)9-s + 0.438·10-s + (−0.5 − 0.559i)12-s + 0.674·13-s − 0.153i·14-s + (0.653 − 0.584i)15-s + 0.312·16-s + 0.352i·17-s + (0.496 + 0.0555i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.64046 - 0.733639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64046 - 0.733639i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2 + 2.23i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + iT - 4T^{2} \) |
| 5 | \( 1 - 4.38iT - 25T^{2} \) |
| 7 | \( 1 - 2.14T + 49T^{2} \) |
| 13 | \( 1 - 8.76T + 169T^{2} \) |
| 17 | \( 1 - 6iT - 289T^{2} \) |
| 19 | \( 1 - 31.1T + 361T^{2} \) |
| 23 | \( 1 - 26.4iT - 529T^{2} \) |
| 29 | \( 1 + 47.3iT - 841T^{2} \) |
| 31 | \( 1 + 30.7T + 961T^{2} \) |
| 37 | \( 1 - 29.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 39.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 4.29iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 27.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 88.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 61.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.66iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100.T + 9.40e3T^{2} \) |
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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
−11.30166425994109625271539832461, −10.56687476745203588350543010056, −9.559709028086301817883607651376, −7.85789570834334696091431867212, −7.28077651576131790764865755686, −6.32280459242604895288160807167, −5.50504598052909892287120209477, −3.65849268070610119690872428602, −2.43187403562583254810021138253, −1.18976804551589348968445071050,
1.17464158804755337340381810609, 3.20243610403770784419530382343, 4.73092094592789518787482074891, 5.42186447219400716017388317077, 6.39124850655092822046515007479, 7.47059268564355930864237700604, 8.604128853653916092512639934021, 9.408378243265804311344378169867, 10.62094035181890541295352510186, 11.26918834426237851124412810111
