| L(s) = 1 | + (−1.07 − 1.36i)3-s + 2.57·5-s + (−0.703 + 2.91i)9-s − 1.65i·11-s + 5.71i·13-s + (−2.76 − 3.50i)15-s + 7.58·17-s + 2.99i·19-s + 0.287i·23-s + 1.65·25-s + (4.72 − 2.16i)27-s + 2.05i·29-s + 6.01i·31-s + (−2.25 + 1.77i)33-s + 1.75·37-s + ⋯ |
| L(s) = 1 | + (−0.618 − 0.785i)3-s + 1.15·5-s + (−0.234 + 0.972i)9-s − 0.498i·11-s + 1.58i·13-s + (−0.713 − 0.906i)15-s + 1.83·17-s + 0.686i·19-s + 0.0600i·23-s + 0.330·25-s + (0.908 − 0.417i)27-s + 0.382i·29-s + 1.08i·31-s + (−0.391 + 0.308i)33-s + 0.288·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663732166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663732166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.07 + 1.36i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.57T + 5T^{2} \) |
| 11 | \( 1 + 1.65iT - 11T^{2} \) |
| 13 | \( 1 - 5.71iT - 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 19 | \( 1 - 2.99iT - 19T^{2} \) |
| 23 | \( 1 - 0.287iT - 23T^{2} \) |
| 29 | \( 1 - 2.05iT - 29T^{2} \) |
| 31 | \( 1 - 6.01iT - 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 + 0.373T + 47T^{2} \) |
| 53 | \( 1 + 7.77iT - 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 1.02iT - 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.81iT - 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 4.43iT - 97T^{2} \) |
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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
−9.889468486699176995605494563689, −8.977154211758972816460768312506, −8.058489084128506254212793808901, −7.11221931237786244189950243549, −6.34129636532260572581568616855, −5.69227140580058133828547845989, −4.94669039990668550431593926891, −3.46310629857543233901709101171, −2.07365941636855165566956966533, −1.26980433710260186534028489209,
0.916330460113304772180140084591, 2.57341531933965965174112627420, 3.60709209371871961643539658606, 4.85407856974917369159743188596, 5.63538354877199066886134732256, 6.00390867437982024260345905427, 7.25110024546210979182529477150, 8.198815795346348092745342768370, 9.315866001141734206250597446607, 10.01157639695511019970968765246
