L(s) = 1 | + (1.41 + 1.41i)3-s + 1.00i·9-s + (4.24 − 4.24i)11-s + 6·17-s + (−1.41 − 1.41i)19-s + 5i·25-s + (2.82 − 2.82i)27-s + 12·33-s + 6i·41-s + (−7.07 + 7.07i)43-s + 7·49-s + (8.48 + 8.48i)51-s − 4.00i·57-s + (−4.24 + 4.24i)59-s + (−9.89 − 9.89i)67-s + ⋯ |
L(s) = 1 | + (0.816 + 0.816i)3-s + 0.333i·9-s + (1.27 − 1.27i)11-s + 1.45·17-s + (−0.324 − 0.324i)19-s + i·25-s + (0.544 − 0.544i)27-s + 2.08·33-s + 0.937i·41-s + (−1.07 + 1.07i)43-s + 49-s + (1.18 + 1.18i)51-s − 0.529i·57-s + (−0.552 + 0.552i)59-s + (−1.20 − 1.20i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.327185142\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.327185142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1.41 - 1.41i)T + 3iT^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (-4.24 + 4.24i)T - 11iT^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (1.41 + 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (7.07 - 7.07i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + (4.24 - 4.24i)T - 59iT^{2} \) |
| 61 | \( 1 + 61iT^{2} \) |
| 67 | \( 1 + (9.89 + 9.89i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (12.7 + 12.7i)T + 83iT^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.782019503702396410359654986628, −9.182670432242331761200101599653, −8.557472666410366965097420416592, −7.72227615398824453773489473521, −6.53437008958181231624385089661, −5.71636258565650270269463030084, −4.52360019677380142021026956134, −3.54031971303229799719004923153, −3.05685879254887893456364875774, −1.24426970780453296550600336089,
1.36316755794293251778440276709, 2.26101314476123488309729218949, 3.49806419544197140533943185756, 4.48007519787570281915130740581, 5.71871041225576753810087099064, 6.83411441455717146374666129443, 7.33448251455460937205347965085, 8.214024324712599547107324302608, 8.924770306307159594812659117154, 9.837442504727420450991999593486