| L(s) = 1 | + (−1.55 + 1.26i)2-s + (0.815 − 3.91i)4-s + 2.61·5-s + 10.9i·7-s + (3.67 + 7.10i)8-s + (−4.05 + 3.29i)10-s + 21.8i·11-s − 4.95·13-s + (−13.8 − 17.0i)14-s + (−14.6 − 6.38i)16-s + 20.1·17-s − 4.35i·19-s + (2.13 − 10.2i)20-s + (−27.5 − 33.9i)22-s − 9.15i·23-s + ⋯ |
| L(s) = 1 | + (−0.775 + 0.630i)2-s + (0.203 − 0.978i)4-s + 0.522·5-s + 1.57i·7-s + (0.459 + 0.888i)8-s + (−0.405 + 0.329i)10-s + 1.98i·11-s − 0.381·13-s + (−0.990 − 1.21i)14-s + (−0.916 − 0.399i)16-s + 1.18·17-s − 0.229i·19-s + (0.106 − 0.511i)20-s + (−1.25 − 1.54i)22-s − 0.398i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9889469486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9889469486\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.55 - 1.26i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 2.61T + 25T^{2} \) |
| 7 | \( 1 - 10.9iT - 49T^{2} \) |
| 11 | \( 1 - 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 4.95T + 169T^{2} \) |
| 17 | \( 1 - 20.1T + 289T^{2} \) |
| 23 | \( 1 + 9.15iT - 529T^{2} \) |
| 29 | \( 1 - 1.24T + 841T^{2} \) |
| 31 | \( 1 + 39.9iT - 961T^{2} \) |
| 37 | \( 1 + 56.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 80.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 16.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 21.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 74.7T + 9.40e3T^{2} \) |
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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
−10.19852264067282292555318308208, −9.653210515332860507681780283409, −9.146053460398963646981768426055, −8.034396730292817404106908948230, −7.30841084782582624609183199441, −6.23334560971151211595083018220, −5.49645331612434199151531846386, −4.65481588658531396354675319026, −2.52428218492826393428675608829, −1.74730558069367980826209196210,
0.46347072529791583861496333690, 1.49877911887050526174728820960, 3.21527869250760521774543557153, 3.77397974230371492739668435690, 5.33746634409913282290386975663, 6.52128103467278195979977884663, 7.48062301038601915320313759840, 8.199235222783436896812760109355, 9.113414256276034163206741875200, 10.15270601644934900419385665350
