L(s) = 1 | − 3-s + (1 + 2i)5-s + 2i·7-s + 9-s − 2i·11-s + 2·13-s + (−1 − 2i)15-s − 6i·17-s − 8i·19-s − 2i·21-s − 4i·23-s + (−3 + 4i)25-s − 27-s + 8i·29-s + 2i·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (0.447 + 0.894i)5-s + 0.755i·7-s + 0.333·9-s − 0.603i·11-s + 0.554·13-s + (−0.258 − 0.516i)15-s − 1.45i·17-s − 1.83i·19-s − 0.436i·21-s − 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s + 1.48i·29-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6820523762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6820523762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.449617658935682980101228900197, −7.13189537479291351882587072422, −6.85242489540295407919268452300, −6.09899961831971042395416080145, −5.25039049660556180526269115043, −4.78381778627784541015896606327, −3.28607537413914765863373658962, −2.84451697234790793650844228161, −1.70187213293398571956481567266, −0.20946061459046152535788349015,
1.37645479050869831430177493519, 1.78773574803789198224948813380, 3.61461112464794846180438201666, 4.11091320673838646542904295556, 4.98719865974542257406460526183, 5.85395243289108653291132987370, 6.23523162061230787235868153417, 7.26665600984162297707491558447, 8.054583092796113678871690138821, 8.561753271073022128325897227349