L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s − 1.41i·6-s + 7-s + 1.00·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 17-s + 1.41i·19-s − 21-s + 1.41i·23-s + 25-s + 2.00·26-s + 27-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s − 1.41i·6-s + 7-s + 1.00·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 17-s + 1.41i·19-s − 21-s + 1.41i·23-s + 25-s + 2.00·26-s + 27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7099385350\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7099385350\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2243 | \( 1+O(T) \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.357508518898910300656862896331, −8.398170831540644111256668012385, −8.028405585152080278133184181256, −7.18820229746084786660285986018, −6.45226372269689607501465298815, −5.54160177370721695696376797694, −5.33740031722516181176432180321, −4.51918019933788515608940550875, −3.12727354174716180371909470310, −1.49715501192150369592956069868,
0.56997466504479186504109130291, 1.96616399236372515750404622028, 2.58769562199968904272912003106, 4.12953025014951936411201534367, 4.55728277366969868345862911955, 5.37567691996034705903213635450, 6.65827870808607036557867147840, 6.94352596521534409344545396053, 8.482199922414724606848310019976, 8.962393933796387657365311424546