L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.189 − 2.63i)7-s − 0.999·8-s + (4.67 − 2.69i)11-s + 2.51·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (3.89 − 2.24i)17-s + (2.48 + 1.43i)19-s − 5.39i·22-s + (−0.133 + 0.232i)23-s + (1.25 − 2.18i)26-s + (−2.38 + 1.15i)28-s + 8.89i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0716 − 0.997i)7-s − 0.353·8-s + (1.40 − 0.813i)11-s + 0.698·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.945 − 0.545i)17-s + (0.568 + 0.328i)19-s − 1.15i·22-s + (−0.0279 + 0.0483i)23-s + (0.246 − 0.427i)26-s + (−0.449 + 0.218i)28-s + 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.643994598\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.643994598\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.189 + 2.63i)T \) |
good | 11 | \( 1 + (-4.67 + 2.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + (-3.89 + 2.24i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.48 - 1.43i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.133 - 0.232i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.64 - 3.25i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.760T + 41T^{2} \) |
| 43 | \( 1 - 5.86iT - 43T^{2} \) |
| 47 | \( 1 + (6.92 + 3.99i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.19 + 7.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.33 - 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.50 + 4.91i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.29 + 7.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.45iT - 83T^{2} \) |
| 89 | \( 1 + (3.98 - 6.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−8.514877375872339653587954950522, −7.79423598372321039467031495085, −6.79453857665354795760684845361, −6.23134566078042535241586766132, −5.28945671914877140630662273748, −4.41270314709826897565398110798, −3.54480020731060863020321754493, −3.16184827610681392883344713605, −1.46867500971085793788305697544, −0.904774314976272577047404152424,
1.25598477493445733289052366314, 2.43840586711803837661073072675, 3.54663797326827718811255455170, 4.26858396448628757800688884560, 5.15781729629869107002497149106, 6.03504985438182233190722955758, 6.41529542749222802636390782331, 7.36474761706238316500065013597, 8.147738832635390790848293384570, 8.765476666692740575353369567323