L(s) = 1 | + (1.22 + 0.707i)2-s + (0.707 − 0.707i)3-s + (0.999 + 1.73i)4-s + (1.36 − 0.366i)6-s + (1.74 + 1.74i)7-s + 2.82i·8-s − 1.00i·9-s − 2i·11-s + (1.93 + 0.517i)12-s + (−4.05 − 4.05i)13-s + (0.901 + 3.36i)14-s + (−2.00 + 3.46i)16-s + (−4.24 + 4.24i)17-s + (0.707 − 1.22i)18-s + 7.19·19-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)2-s + (0.408 − 0.408i)3-s + (0.499 + 0.866i)4-s + (0.557 − 0.149i)6-s + (0.658 + 0.658i)7-s + 0.999i·8-s − 0.333i·9-s − 0.603i·11-s + (0.557 + 0.149i)12-s + (−1.12 − 1.12i)13-s + (0.241 + 0.899i)14-s + (−0.500 + 0.866i)16-s + (−1.02 + 1.02i)17-s + (0.166 − 0.288i)18-s + 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23966 + 0.760570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23966 + 0.760570i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.74 - 1.74i)T + 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (4.05 + 4.05i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + (-0.378 + 0.378i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.46iT - 29T^{2} \) |
| 31 | \( 1 + 0.267iT - 31T^{2} \) |
| 37 | \( 1 + (2.07 - 2.07i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + (1.74 - 1.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (9.14 + 9.14i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.03 + 1.03i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.53T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 + (-8.81 - 8.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.46iT - 71T^{2} \) |
| 73 | \( 1 + (-2.82 - 2.82i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 + (-0.656 + 0.656i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (-4.43 + 4.43i)T - 97iT^{2} \) |
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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
−11.95377649411396589969668001742, −11.35856119201973120324605402910, −9.945251920964549828444733783536, −8.497948413998902741354350697817, −8.012734528146882933106141967078, −6.94224559173018258728504272135, −5.75924135752740137949203917693, −4.93455500706823304270614398985, −3.41108022194449428665309711009, −2.25269936065356816363764482463,
1.83813186662703678092373766035, 3.22439725721287402853970558972, 4.63842603732416048319774398786, 4.97637969915031303869463374087, 6.85390319966154617385205353610, 7.48718051001369182114759195190, 9.226831197832524169545430055094, 9.796209540641302920601591240297, 10.92664802447373699603918104285, 11.62296505773825312123413379921