| L(s) = 1 | + 1.39·2-s − 2.89·3-s − 0.0576·4-s − 2.32·5-s − 4.03·6-s − 2.86·8-s + 5.37·9-s − 3.24·10-s + 3.46·11-s + 0.166·12-s − 1.96·13-s + 6.73·15-s − 3.88·16-s + 0.123·17-s + 7.49·18-s − 4.79·19-s + 0.134·20-s + 4.82·22-s − 2.03·23-s + 8.29·24-s + 0.414·25-s − 2.74·26-s − 6.87·27-s + 5.20·29-s + 9.38·30-s − 0.637·31-s + 0.325·32-s + ⋯ |
| L(s) = 1 | + 0.985·2-s − 1.67·3-s − 0.0288·4-s − 1.04·5-s − 1.64·6-s − 1.01·8-s + 1.79·9-s − 1.02·10-s + 1.04·11-s + 0.0481·12-s − 0.545·13-s + 1.73·15-s − 0.970·16-s + 0.0298·17-s + 1.76·18-s − 1.10·19-s + 0.0299·20-s + 1.02·22-s − 0.425·23-s + 1.69·24-s + 0.0828·25-s − 0.537·26-s − 1.32·27-s + 0.965·29-s + 1.71·30-s − 0.114·31-s + 0.0576·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 - 0.123T + 17T^{2} \) |
| 19 | \( 1 + 4.79T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 0.637T + 31T^{2} \) |
| 37 | \( 1 + 4.00T + 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.121T + 53T^{2} \) |
| 59 | \( 1 - 4.93T + 59T^{2} \) |
| 61 | \( 1 - 9.06T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.30687222518560708280020036858, −6.79221036180544278402637147414, −5.99190079490766379225099093988, −5.64184648714383633132878793816, −4.61569031695154726823166394108, −4.26705388622121927173847987521, −3.77536164008587284126493728050, −2.48008842646177918807343345874, −0.930809305792159985559736949565, 0,
0.930809305792159985559736949565, 2.48008842646177918807343345874, 3.77536164008587284126493728050, 4.26705388622121927173847987521, 4.61569031695154726823166394108, 5.64184648714383633132878793816, 5.99190079490766379225099093988, 6.79221036180544278402637147414, 7.30687222518560708280020036858
