L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.884 − 2.49i)7-s + 1.00i·9-s + 3.27·11-s + (−3.02 − 3.02i)13-s + (1.41 − 1.41i)17-s − 2.15·19-s + (−1.13 + 2.38i)21-s + (0.296 − 0.296i)23-s + (0.707 − 0.707i)27-s − 0.725i·29-s + 1.31i·31-s + (−2.31 − 2.31i)33-s + (−1.56 − 1.56i)37-s + 4.27i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.334 − 0.942i)7-s + 0.333i·9-s + 0.987·11-s + (−0.838 − 0.838i)13-s + (0.342 − 0.342i)17-s − 0.493·19-s + (−0.248 + 0.521i)21-s + (0.0617 − 0.0617i)23-s + (0.136 − 0.136i)27-s − 0.134i·29-s + 0.235i·31-s + (−0.403 − 0.403i)33-s + (−0.256 − 0.256i)37-s + 0.684i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7472824332\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7472824332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.884 + 2.49i)T \) |
good | 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + (3.02 + 3.02i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 + (-0.296 + 0.296i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.725iT - 29T^{2} \) |
| 31 | \( 1 - 1.31iT - 31T^{2} \) |
| 37 | \( 1 + (1.56 + 1.56i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.25iT - 41T^{2} \) |
| 43 | \( 1 + (0.632 - 0.632i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.71 - 3.71i)T - 53iT^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (-3.08 - 3.08i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + (7.07 + 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (11.7 + 11.7i)T + 83iT^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \) |
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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
−8.724887389814076440232008912695, −7.74612624232029558273161211290, −7.17142168355374583580602742902, −6.50046628710744093198954823288, −5.63623567767934507849771468756, −4.69352941588145169392093209117, −3.82584040489245370060071853574, −2.80118499899797799301370477302, −1.44012719567720811261460425336, −0.28360404405028755546464028015,
1.58946476737879117174633165553, 2.73892021140175452951243735071, 3.82649494748603326633887807739, 4.64630316735803631555187796007, 5.50960130825219107576139171466, 6.35780653933848301265190306918, 6.86524382642475294877887402634, 8.026680850001413414312352594683, 8.896773421027328315728588972101, 9.468330433754720294242887321335