L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s − 37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s − 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3735359401\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3735359401\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−15.78158863451418310600567398366, −13.91118790105346893717398966072, −13.04786003793567616204031954427, −12.40570419368847873598634726769, −11.16295984834053954222603330510, −9.530714621167639954748207724266, −8.113025414063899120618495471630, −7.02046440325267567096761130699, −5.50846861495591810506841875830, −3.30843370639598070011740501216,
3.89339273183383705579882973780, 5.42600079494021974315876790764, 6.56213261943062567216093815027, 8.979906840158385872094533206307, 9.755934096435888961389015983965, 10.73867704030288327198980522926, 12.01208341174308721549737477039, 13.52036331503366893986103235457, 14.50422369454586197462514135495, 15.73698933792598545940443005947