L(s) = 1 | + (−0.105 + 0.394i)2-s + (0.965 − 0.258i)3-s + (1.58 + 0.916i)4-s + 0.408i·6-s + (−2.57 + 0.605i)7-s + (−1.10 + 1.10i)8-s + (0.866 − 0.499i)9-s + (−0.463 + 0.803i)11-s + (1.77 + 0.474i)12-s + (4.08 + 4.08i)13-s + (0.0332 − 1.08i)14-s + (1.51 + 2.62i)16-s + (0.192 + 0.719i)17-s + (0.105 + 0.394i)18-s + (1.21 + 2.11i)19-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.278i)2-s + (0.557 − 0.149i)3-s + (0.793 + 0.458i)4-s + 0.166i·6-s + (−0.973 + 0.228i)7-s + (−0.391 + 0.391i)8-s + (0.288 − 0.166i)9-s + (−0.139 + 0.242i)11-s + (0.511 + 0.136i)12-s + (1.13 + 1.13i)13-s + (0.00889 − 0.288i)14-s + (0.378 + 0.655i)16-s + (0.0467 + 0.174i)17-s + (0.0249 + 0.0929i)18-s + (0.279 + 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56247 + 0.972166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56247 + 0.972166i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.57 - 0.605i)T \) |
good | 2 | \( 1 + (0.105 - 0.394i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (0.463 - 0.803i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.08 - 4.08i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.192 - 0.719i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.21 - 2.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.00 - 1.34i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 8.08iT - 29T^{2} \) |
| 31 | \( 1 + (1.05 + 0.607i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.472 + 1.76i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 6.97iT - 41T^{2} \) |
| 43 | \( 1 + (-0.781 + 0.781i)T - 43iT^{2} \) |
| 47 | \( 1 + (10.0 + 2.70i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.72 + 6.42i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.91 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.72 - 2.15i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 2.80i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + (4.02 - 1.07i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (7.02 - 4.05i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.91 + 5.91i)T + 83iT^{2} \) |
| 89 | \( 1 + (7.78 + 13.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \) |
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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
−11.13821991767497258230442954576, −9.991932508762966450068267215827, −9.094861328306913683110470650838, −8.345129619220698700741811195177, −7.31459954651442800154088544210, −6.59530203471287086365354462603, −5.79419930856181416961845892870, −3.99789040299615218373202952743, −3.11273869244145091064215599576, −1.89912858962416989162734571039,
1.12982273537291892939332138098, 2.92654357934767694717294767891, 3.36843209817648434593990974304, 5.11993145054792464928327820337, 6.23588892590469859217209574629, 6.97501314584360784061902862757, 8.060803482059470977928639915478, 9.116027840236282773632074017866, 9.875954556502251332225377126397, 10.75830026674331646433900051178