| L(s) = 1 | + 0.483·3-s − 0.806·5-s − 2.98·7-s − 2.76·9-s − 0.460·11-s + 0.743·13-s − 0.389·15-s + 6.59·17-s − 3.07·19-s − 1.44·21-s + 4.61·23-s − 4.34·25-s − 2.78·27-s + 8.64·29-s + 4.60·31-s − 0.222·33-s + 2.40·35-s + 7.56·37-s + 0.359·39-s − 0.769·41-s + 3.20·43-s + 2.23·45-s + 47-s + 1.91·49-s + 3.18·51-s − 7.24·53-s + 0.371·55-s + ⋯ |
| L(s) = 1 | + 0.279·3-s − 0.360·5-s − 1.12·7-s − 0.922·9-s − 0.138·11-s + 0.206·13-s − 0.100·15-s + 1.59·17-s − 0.706·19-s − 0.314·21-s + 0.963·23-s − 0.869·25-s − 0.536·27-s + 1.60·29-s + 0.827·31-s − 0.0387·33-s + 0.407·35-s + 1.24·37-s + 0.0575·39-s − 0.120·41-s + 0.489·43-s + 0.332·45-s + 0.145·47-s + 0.273·49-s + 0.446·51-s − 0.995·53-s + 0.0501·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 - 0.483T + 3T^{2} \) |
| 5 | \( 1 + 0.806T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 + 0.460T + 11T^{2} \) |
| 13 | \( 1 - 0.743T + 13T^{2} \) |
| 17 | \( 1 - 6.59T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 8.64T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 + 0.769T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 53 | \( 1 + 7.24T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 0.940T + 61T^{2} \) |
| 67 | \( 1 + 8.64T + 67T^{2} \) |
| 71 | \( 1 - 0.347T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 0.523T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−7.960851148662152613146637106693, −7.00577105562483913182625369286, −6.19512373899045549084420101816, −5.80711419150776215670880051796, −4.77958441443500954165954872435, −3.91383641627528865792915833993, −2.98334817803306210468119060549, −2.79297253923280381835210491211, −1.19170482331845938424090839150, 0,
1.19170482331845938424090839150, 2.79297253923280381835210491211, 2.98334817803306210468119060549, 3.91383641627528865792915833993, 4.77958441443500954165954872435, 5.80711419150776215670880051796, 6.19512373899045549084420101816, 7.00577105562483913182625369286, 7.960851148662152613146637106693
