| L(s) = 1 | + (−0.358 − 2.97i)3-s + 0.408i·5-s + (−0.674 + 1.16i)7-s + (−8.74 + 2.13i)9-s + (16.2 + 9.38i)11-s + (3.16 − 5.47i)13-s + (1.21 − 0.146i)15-s + (1.03 + 0.599i)17-s + (−4.34 − 18.4i)19-s + (3.72 + 1.58i)21-s + (15.2 + 8.79i)23-s + 24.8·25-s + (9.50 + 25.2i)27-s − 18.6i·29-s + (16.2 + 28.0i)31-s + ⋯ |
| L(s) = 1 | + (−0.119 − 0.992i)3-s + 0.0816i·5-s + (−0.0963 + 0.166i)7-s + (−0.971 + 0.237i)9-s + (1.47 + 0.853i)11-s + (0.243 − 0.421i)13-s + (0.0810 − 0.00977i)15-s + (0.0610 + 0.0352i)17-s + (−0.228 − 0.973i)19-s + (0.177 + 0.0756i)21-s + (0.662 + 0.382i)23-s + 0.993·25-s + (0.352 + 0.935i)27-s − 0.641i·29-s + (0.523 + 0.906i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.819790871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819790871\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.358 + 2.97i)T \) |
| 19 | \( 1 + (4.34 + 18.4i)T \) |
| good | 5 | \( 1 - 0.408iT - 25T^{2} \) |
| 7 | \( 1 + (0.674 - 1.16i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-16.2 - 9.38i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.16 + 5.47i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 0.599i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-15.2 - 8.79i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 18.6iT - 841T^{2} \) |
| 31 | \( 1 + (-16.2 - 28.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 18.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (33.5 + 58.1i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 72.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-5.40 + 3.11i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 46.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (27.5 - 47.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-48.7 - 28.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (27.9 - 48.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (39.1 + 67.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (57.7 + 33.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-130. + 75.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-37.2 - 64.5i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.16189471644821086682067797402, −9.024112583354543198112943793866, −8.520002351299206778007784846119, −7.12240035300391745861903237962, −6.87779610137546803539810643168, −5.79932008403711273997970415376, −4.69340021323142122821873894507, −3.33003251309818700824399432557, −2.08047774918031019593647463813, −0.853483626993287707195758733763,
1.10037480908847472538294398093, 3.02873717284814096442431547566, 3.93886422009199305448921578103, 4.76233049245989011565373599418, 6.03032915170647858294283127499, 6.58103779125887146800767442836, 8.073132118546068172589067409236, 8.911448513843107576934802174357, 9.477528568760990493617395462473, 10.43381195628854311167837205679
