| L(s) = 1 | + 3i·3-s + 2·7-s − 9·9-s − 22i·13-s − 26i·19-s + 6i·21-s − 25·25-s − 27i·27-s + 46·31-s − 26i·37-s + 66·39-s − 22i·43-s − 45·49-s + 78·57-s + 74i·61-s + ⋯ |
| L(s) = 1 | + i·3-s + 0.285·7-s − 9-s − 1.69i·13-s − 1.36i·19-s + 0.285i·21-s − 25-s − i·27-s + 1.48·31-s − 0.702i·37-s + 1.69·39-s − 0.511i·43-s − 0.918·49-s + 1.36·57-s + 1.21i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.366108919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366108919\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 22iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 26iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 46T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 22iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 74iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 122iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 142T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−10.07894317476087984943378884364, −9.267560219003402184536503683054, −8.359960403707787313164502067115, −7.65643747070575351854773007716, −6.30416131674938740640170412013, −5.36963711045707406624635777561, −4.64603999232753337513652077992, −3.49077546506200036155167067615, −2.54094146412714703017665163071, −0.48709388338225333074117870277,
1.34919654385142630522390287015, 2.26818724145147922183916076725, 3.70784312821415518951851250659, 4.86336699844521943970010196486, 6.11170861738083795593455446879, 6.64362257394507595800497411810, 7.72813444060637483054792909800, 8.310756588004150683219898105583, 9.298124431561620862933162557262, 10.18001498609271687734124239396
