L(s) = 1 | + (0.618 + 1.61i)3-s + (1 + i)7-s + (−2.23 + 2.00i)9-s + 4.47i·11-s + (3 − 3i)13-s + (−2.23 + 2.23i)17-s − 2i·19-s + (−1 + 2.23i)21-s + (2.23 + 2.23i)23-s + (−4.61 − 2.38i)27-s − 4.47·29-s + 4·31-s + (−7.23 + 2.76i)33-s + (3 + 3i)37-s + (6.70 + 3i)39-s + ⋯ |
L(s) = 1 | + (0.356 + 0.934i)3-s + (0.377 + 0.377i)7-s + (−0.745 + 0.666i)9-s + 1.34i·11-s + (0.832 − 0.832i)13-s + (−0.542 + 0.542i)17-s − 0.458i·19-s + (−0.218 + 0.487i)21-s + (0.466 + 0.466i)23-s + (−0.888 − 0.458i)27-s − 0.830·29-s + 0.718·31-s + (−1.25 + 0.481i)33-s + (0.493 + 0.493i)37-s + (1.07 + 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11632 + 0.923303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11632 + 0.923303i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.23 - 2.23i)T - 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.23 + 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (6.70 + 6.70i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + (9 + 9i)T + 97iT^{2} \) |
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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
−11.79393281739282728100547967585, −10.85166698912862104749893136814, −10.11426050881892271880340441366, −9.109105243283859781380181217322, −8.361699605012605427135800660692, −7.24355413676920567416071509382, −5.75087394928268454706433974125, −4.78111411192609383689351198856, −3.69176098081617736417805070568, −2.22398376156770181511611763688,
1.16587846828609069376831781271, 2.81455434271814746271916508568, 4.15978367529965748139362840761, 5.82979013310418854009129207376, 6.64238890294253049764031788464, 7.75805406001279014910180815952, 8.569981110444278696405843356746, 9.375542861506582443292378434625, 11.04353774483908957040979734154, 11.35391331091906233561861649643