| L(s) = 1 | − 2-s + 2.30·3-s + 4-s − 2.30·6-s + 0.697·7-s − 8-s + 2.30·9-s − 2.60·11-s + 2.30·12-s − 0.605·13-s − 0.697·14-s + 16-s + 2.60·17-s − 2.30·18-s + 5.90·19-s + 1.60·21-s + 2.60·22-s − 6.90·23-s − 2.30·24-s − 5·25-s + 0.605·26-s − 1.60·27-s + 0.697·28-s − 7.30·29-s − 2.30·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.940·6-s + 0.263·7-s − 0.353·8-s + 0.767·9-s − 0.785·11-s + 0.664·12-s − 0.167·13-s − 0.186·14-s + 0.250·16-s + 0.631·17-s − 0.542·18-s + 1.35·19-s + 0.350·21-s + 0.555·22-s − 1.44·23-s − 0.470·24-s − 25-s + 0.118·26-s − 0.308·27-s + 0.131·28-s − 1.35·29-s − 0.413·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120096950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120096950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 - 0.302T + 37T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 - 9.90T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−13.71962328036536330951832520373, −12.44326425251779586987482344508, −11.22995683601162525762162664520, −9.911769729603112665526743495407, −9.254739625139299623182027251413, −7.908480173217457247900990353429, −7.62946148734449235936311625909, −5.64098251036911842491528267321, −3.59192158714879233148290807179, −2.17613306455745034996185593230,
2.17613306455745034996185593230, 3.59192158714879233148290807179, 5.64098251036911842491528267321, 7.62946148734449235936311625909, 7.908480173217457247900990353429, 9.254739625139299623182027251413, 9.911769729603112665526743495407, 11.22995683601162525762162664520, 12.44326425251779586987482344508, 13.71962328036536330951832520373
