| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 1.48·5-s + 9.64·7-s − 2.82i·8-s + 2.10i·10-s + 3.77i·11-s + 23.9·13-s + 13.6i·14-s + 4.00·16-s − 14.9·17-s + 21.1i·19-s − 2.97·20-s − 5.33·22-s + 29.3·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.297·5-s + 1.37·7-s − 0.353i·8-s + 0.210i·10-s + 0.342i·11-s + 1.83·13-s + 0.973i·14-s + 0.250·16-s − 0.880·17-s + 1.11i·19-s − 0.148·20-s − 0.242·22-s + 1.27·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0968 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0968 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.689624087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.689624087\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 113 | \( 1 + (-56.0 - 98.1i)T \) |
| good | 5 | \( 1 - 1.48T + 25T^{2} \) |
| 7 | \( 1 - 9.64T + 49T^{2} \) |
| 11 | \( 1 - 3.77iT - 121T^{2} \) |
| 13 | \( 1 - 23.9T + 169T^{2} \) |
| 17 | \( 1 + 14.9T + 289T^{2} \) |
| 19 | \( 1 - 21.1iT - 361T^{2} \) |
| 23 | \( 1 - 29.3T + 529T^{2} \) |
| 29 | \( 1 + 47.6T + 841T^{2} \) |
| 31 | \( 1 - 41.4T + 961T^{2} \) |
| 37 | \( 1 - 69.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 4.82iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 101. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 7.38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 78.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 57.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 63.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 68.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 27.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 35.5T + 9.40e3T^{2} \) |
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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
−8.771083843637791642086700589773, −8.414490715331519038216503596635, −7.66247146247835939141468033879, −6.76022115513430860983048812621, −5.95459116929924025648845248561, −5.26567389440587185612544716223, −4.37152805953842016416775570437, −3.59529125276492891635873252596, −1.99922734185160900081508137718, −1.13656639614824012137679847564,
0.804219614944832875607307271825, 1.70313852182630885590569322095, 2.67116919323198111814778632372, 3.86567258014670997941462169609, 4.54995506392450208886273984419, 5.49627327824793717419405165100, 6.22046972274542095247038131788, 7.39461614566311163024466287616, 8.171358404757238285914200179746, 9.035653976093266287582558449742
