L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.06 − 1.96i)5-s + (3.58 + 3.58i)7-s + (−0.707 − 0.707i)8-s + (−0.632 − 2.14i)10-s − 4.10i·11-s + (−1.38 + 1.38i)13-s + 5.07·14-s − 1.00·16-s + (1.61 − 1.61i)17-s + i·19-s + (−1.96 − 1.06i)20-s + (−2.90 − 2.90i)22-s + (−4.18 − 4.18i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.478 − 0.878i)5-s + (1.35 + 1.35i)7-s + (−0.250 − 0.250i)8-s + (−0.199 − 0.678i)10-s − 1.23i·11-s + (−0.384 + 0.384i)13-s + 1.35·14-s − 0.250·16-s + (0.392 − 0.392i)17-s + 0.229i·19-s + (−0.439 − 0.239i)20-s + (−0.618 − 0.618i)22-s + (−0.871 − 0.871i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.846576618\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.846576618\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.06 + 1.96i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-3.58 - 3.58i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.10iT - 11T^{2} \) |
| 13 | \( 1 + (1.38 - 1.38i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.61 + 1.61i)T - 17iT^{2} \) |
| 23 | \( 1 + (4.18 + 4.18i)T + 23iT^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 + (1.38 + 1.38i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.96iT - 41T^{2} \) |
| 43 | \( 1 + (-5.42 + 5.42i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.09 - 7.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.514 - 0.514i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.814T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 3.04iT - 71T^{2} \) |
| 73 | \( 1 + (-6.42 + 6.42i)T - 73iT^{2} \) |
| 79 | \( 1 + 10.9iT - 79T^{2} \) |
| 83 | \( 1 + (8.82 + 8.82i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.88T + 89T^{2} \) |
| 97 | \( 1 + (7.03 + 7.03i)T + 97iT^{2} \) |
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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
−9.059920635649304019140882068181, −8.461102968889537741934645228923, −7.968860093834553021051021549094, −6.27931207175696269187636627020, −5.80826754939116132863338477941, −4.89712012313644611473146153572, −4.48617469223683721909178060323, −2.91493297725897178287621318321, −2.11593057956916581847892371147, −1.02723472786398099532833521522,
1.45502944244927230499592771353, 2.59271311687310957772829743439, 3.83885285607286112846026155759, 4.60129397199001450079018016952, 5.30416169755039681231658927309, 6.47153531584511959753282504502, 7.08802687211858789884045071452, 7.77001662481805924398317722675, 8.284506596743684563557456248432, 9.881340923968266931564097358574