L(s) = 1 | + (1.36 − 0.366i)2-s + (1.73 − i)4-s − 1.12·5-s + i·7-s + (1.99 − 2i)8-s + (−1.54 + 0.413i)10-s − 3.86i·11-s + 5.08i·13-s + (0.366 + 1.36i)14-s + (1.99 − 3.46i)16-s − 2i·17-s + 7.99·19-s + (−1.95 + 1.12i)20-s + (−1.41 − 5.27i)22-s + 7.68·23-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s − 0.505·5-s + 0.377i·7-s + (0.707 − 0.707i)8-s + (−0.488 + 0.130i)10-s − 1.16i·11-s + 1.41i·13-s + (0.0978 + 0.365i)14-s + (0.499 − 0.866i)16-s − 0.485i·17-s + 1.83·19-s + (−0.437 + 0.252i)20-s + (−0.301 − 1.12i)22-s + 1.60·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.087901571\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.087901571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 1.12T + 5T^{2} \) |
| 11 | \( 1 + 3.86iT - 11T^{2} \) |
| 13 | \( 1 - 5.08iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 7.99T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 4.08iT - 31T^{2} \) |
| 37 | \( 1 + 8.28iT - 37T^{2} \) |
| 41 | \( 1 + 8.41iT - 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 - 7.72T + 53T^{2} \) |
| 59 | \( 1 - 10.9iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 5.08T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.63T + 73T^{2} \) |
| 79 | \( 1 - 7.17iT - 79T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.17T + 97T^{2} \) |
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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
−9.345022191103800518688349557127, −8.698931975815026066380791689500, −7.42980770727861736458941524874, −6.96856445409531789834215083084, −5.84467686062595664036046438431, −5.25073789592091124198759351518, −4.21040677457172613559844051990, −3.39316964897096732452893671408, −2.50418834702289209286596715828, −1.04651832671260405631779705897,
1.35538471196121924546206470057, 2.96740213633203243321329636370, 3.49332317027677041487147603625, 4.80611508928106236991808774188, 5.10914515814431260585737884721, 6.33199091610995526098759331654, 7.17322285946874360868703462483, 7.72353810115171926569197102699, 8.437117777337960570838896415822, 9.828223875926158508865446202802