L(s) = 1 | + (1.22 − 1.22i)2-s − 1.99i·4-s + (−1.22 − 1.22i)8-s − i·9-s − 11-s − 0.999·16-s + (−1.22 − 1.22i)18-s + (−1.22 + 1.22i)22-s + (1.22 + 1.22i)23-s − i·29-s − 1.99·36-s + (−1.22 + 1.22i)37-s + (1.22 + 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯ |
L(s) = 1 | + (1.22 − 1.22i)2-s − 1.99i·4-s + (−1.22 − 1.22i)8-s − i·9-s − 11-s − 0.999·16-s + (−1.22 − 1.22i)18-s + (−1.22 + 1.22i)22-s + (1.22 + 1.22i)23-s − i·29-s − 1.99·36-s + (−1.22 + 1.22i)37-s + (1.22 + 1.22i)43-s + 1.99i·44-s + 2.99·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.881170406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.881170406\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.875843711614699068623930413434, −9.214534471159381407180545567801, −8.020049059268566005772745327633, −6.91240748978798720574315366460, −5.89244075884852514837005178803, −5.20173850773963351762683898968, −4.28739158381938449392032060958, −3.33972925965622599452415660909, −2.64888801856488417467285598299, −1.26785764692904594031077648994,
2.37978861687901926380447512796, 3.41331898879863638145969759964, 4.60297917776591991362979888643, 5.14265728771490066099987508288, 5.84404067243340465214747941895, 6.95859991922082585241453399050, 7.43697108886134340945705141768, 8.276299641931899120166096716015, 9.005042311073999745563272096365, 10.52898966185803739289781108359