| L(s) = 1 | − 0.482·2-s − 0.300·3-s − 1.76·4-s − 0.848·5-s + 0.144·6-s + 1.74·7-s + 1.81·8-s − 2.90·9-s + 0.409·10-s − 0.172·11-s + 0.530·12-s − 0.843·14-s + 0.254·15-s + 2.65·16-s − 7.81·17-s + 1.40·18-s + 7.12·19-s + 1.49·20-s − 0.524·21-s + 0.0832·22-s + 2.39·23-s − 0.546·24-s − 4.28·25-s + 1.77·27-s − 3.08·28-s + 4.75·29-s − 0.122·30-s + ⋯ |
| L(s) = 1 | − 0.341·2-s − 0.173·3-s − 0.883·4-s − 0.379·5-s + 0.0591·6-s + 0.660·7-s + 0.643·8-s − 0.969·9-s + 0.129·10-s − 0.0519·11-s + 0.153·12-s − 0.225·14-s + 0.0657·15-s + 0.663·16-s − 1.89·17-s + 0.331·18-s + 1.63·19-s + 0.335·20-s − 0.114·21-s + 0.0177·22-s + 0.500·23-s − 0.111·24-s − 0.856·25-s + 0.341·27-s − 0.583·28-s + 0.882·29-s − 0.0224·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.482T + 2T^{2} \) |
| 3 | \( 1 + 0.300T + 3T^{2} \) |
| 5 | \( 1 + 0.848T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 0.172T + 11T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 37 | \( 1 - 0.630T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 - 6.64T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 0.368T + 97T^{2} \) |
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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
−8.033325364151879560004718872376, −7.33413670807894654374918448985, −6.41721931390186435071339052528, −5.49021509053072153926505867279, −4.91144362623610031227414934841, −4.26328900827631793544533335999, −3.35145073025550751619562220312, −2.33320579703391686488646929057, −1.07178957455723841770528542801, 0,
1.07178957455723841770528542801, 2.33320579703391686488646929057, 3.35145073025550751619562220312, 4.26328900827631793544533335999, 4.91144362623610031227414934841, 5.49021509053072153926505867279, 6.41721931390186435071339052528, 7.33413670807894654374918448985, 8.033325364151879560004718872376
