| L(s) = 1 | + 2.05·3-s + 0.680·5-s − 1.59·7-s + 1.23·9-s + 11-s − 0.546·13-s + 1.39·15-s + 6.61·17-s + 19-s − 3.27·21-s + 6.44·23-s − 4.53·25-s − 3.62·27-s + 7.32·29-s − 3.39·31-s + 2.05·33-s − 1.08·35-s + 8.83·37-s − 1.12·39-s + 3.18·41-s − 10.4·43-s + 0.841·45-s + 7.25·47-s − 4.46·49-s + 13.6·51-s + 7.47·53-s + 0.680·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.304·5-s − 0.601·7-s + 0.412·9-s + 0.301·11-s − 0.151·13-s + 0.361·15-s + 1.60·17-s + 0.229·19-s − 0.714·21-s + 1.34·23-s − 0.907·25-s − 0.698·27-s + 1.35·29-s − 0.610·31-s + 0.358·33-s − 0.182·35-s + 1.45·37-s − 0.180·39-s + 0.497·41-s − 1.58·43-s + 0.125·45-s + 1.05·47-s − 0.638·49-s + 1.90·51-s + 1.02·53-s + 0.0916·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.042182614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.042182614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 - 0.680T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 13 | \( 1 + 0.546T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 7.88T + 71T^{2} \) |
| 73 | \( 1 - 3.00T + 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 - 7.76T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 9.99T + 97T^{2} \) |
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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
−8.715260834849848443453776926993, −7.86103183169579208182102254720, −7.38484720463651883372089971254, −6.39284924232815281105256815848, −5.67344047131346778584156176447, −4.71667808142426543505052673762, −3.56543608520028477300660359785, −3.12729999421482292703132001442, −2.24448104569691121917397665888, −1.02374187244259494323978266745,
1.02374187244259494323978266745, 2.24448104569691121917397665888, 3.12729999421482292703132001442, 3.56543608520028477300660359785, 4.71667808142426543505052673762, 5.67344047131346778584156176447, 6.39284924232815281105256815848, 7.38484720463651883372089971254, 7.86103183169579208182102254720, 8.715260834849848443453776926993
