L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−1.03 − 1.98i)5-s + 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (2.13 + 0.674i)10-s + 2.62·11-s + (−0.707 − 0.707i)12-s + (1.21 − 1.21i)13-s + (−2.13 − 0.674i)15-s − 1.00·16-s + (5.35 + 5.35i)17-s + (0.707 + 0.707i)18-s + 4.65·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.460 − 0.887i)5-s + 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.674 + 0.213i)10-s + 0.791·11-s + (−0.204 − 0.204i)12-s + (0.337 − 0.337i)13-s + (−0.550 − 0.174i)15-s − 0.250·16-s + (1.29 + 1.29i)17-s + (0.166 + 0.166i)18-s + 1.06·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.464289717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464289717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.03 + 1.98i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + (-1.21 + 1.21i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.35 - 5.35i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 + (3.62 + 3.62i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.99iT - 29T^{2} \) |
| 31 | \( 1 + 10.0iT - 31T^{2} \) |
| 37 | \( 1 + (-2.79 + 2.79i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.59iT - 41T^{2} \) |
| 43 | \( 1 + (0.545 + 0.545i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.48 - 4.48i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.06 + 6.06i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 + 4.80iT - 61T^{2} \) |
| 67 | \( 1 + (1.81 - 1.81i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 + (-9.65 + 9.65i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.99iT - 79T^{2} \) |
| 83 | \( 1 + (7.99 - 7.99i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.162T + 89T^{2} \) |
| 97 | \( 1 + (4.35 + 4.35i)T + 97iT^{2} \) |
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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
−9.248396307670994699701820671692, −8.395532474380214731162091268432, −7.948902019000844354730174793490, −7.22526804409439759931224649702, −6.09791551621066693508021040917, −5.51348672141917968042845422438, −4.23578070981231827101090955640, −3.43775817702772654436425005535, −1.76017019859077168721287274193, −0.789710201867406458830989221947,
1.25232453525504455818438050351, 2.74443872196109790716486174547, 3.39799753424183149332798411393, 4.20056023666294868707923472834, 5.43159294056038751435766637661, 6.61629269791672570613243976312, 7.43838810182183127974675826572, 8.007752047245774335579930242384, 9.005017735025029712203345760264, 9.783846868209626930840161951024