L(s) = 1 | − 2i·3-s + 1.73i·5-s − i·7-s − 9-s + 5.19·11-s − 4·13-s + 3.46·15-s + 3·17-s + (1.73 + 4i)19-s − 2·21-s + 2.00·25-s − 4i·27-s − 6·29-s + 10.3·31-s − 10.3i·33-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + 0.774i·5-s − 0.377i·7-s − 0.333·9-s + 1.56·11-s − 1.10·13-s + 0.894·15-s + 0.727·17-s + (0.397 + 0.917i)19-s − 0.436·21-s + 0.400·25-s − 0.769i·27-s − 1.11·29-s + 1.86·31-s − 1.80i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.820889175\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820889175\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.73 - 4i)T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−9.818976983066798320017690413751, −8.657406056226757029392131663584, −7.68191606790161535292344126828, −7.09704410691534492595996752076, −6.58615030010862309697059409641, −5.66451625682008719978681713578, −4.29623557514844464208380390629, −3.30091327831325657875194798288, −2.09745859622874046655580551569, −1.01102998948017035303886863168,
1.17134323649567858376662904904, 2.81120755735681013721697386347, 3.95611627419880782726378646121, 4.68759101457801892524595668614, 5.31007649048149705749361548648, 6.44748727415321766850422115321, 7.41933797707272634345819792798, 8.524737632856454398447781502932, 9.349733206506728448831985105173, 9.544681384845321449699929084047