| L(s) = 1 | − 2·4-s + 2·5-s − 7-s + 4·13-s + 4·16-s + 17-s + 7·19-s − 4·20-s − 25-s + 2·28-s + 29-s − 3·31-s − 2·35-s + 8·37-s + 6·41-s + 12·43-s − 8·47-s + 49-s − 8·52-s + 6·53-s − 8·59-s + 13·61-s − 8·64-s + 8·65-s + 67-s − 2·68-s + 11·73-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 0.377·7-s + 1.10·13-s + 16-s + 0.242·17-s + 1.60·19-s − 0.894·20-s − 1/5·25-s + 0.377·28-s + 0.185·29-s − 0.538·31-s − 0.338·35-s + 1.31·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 1.10·52-s + 0.824·53-s − 1.04·59-s + 1.66·61-s − 64-s + 0.992·65-s + 0.122·67-s − 0.242·68-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.470279286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.470279286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.98813812741268, −12.74957792133969, −12.13097628179280, −11.54669222380569, −11.06471716840948, −10.59362659017607, −9.898129542793604, −9.717293417599638, −9.242256912798534, −8.964956537540557, −8.234711164706914, −7.844790240585694, −7.351798755878556, −6.661222882445019, −5.963303527754484, −5.825204590200953, −5.316751271532615, −4.744626811578856, −4.047198844745117, −3.677008515749390, −3.084260360510410, −2.487795498596812, −1.701303076041045, −0.9750333255731493, −0.6599454184035172,
0.6599454184035172, 0.9750333255731493, 1.701303076041045, 2.487795498596812, 3.084260360510410, 3.677008515749390, 4.047198844745117, 4.744626811578856, 5.316751271532615, 5.825204590200953, 5.963303527754484, 6.661222882445019, 7.351798755878556, 7.844790240585694, 8.234711164706914, 8.964956537540557, 9.242256912798534, 9.717293417599638, 9.898129542793604, 10.59362659017607, 11.06471716840948, 11.54669222380569, 12.13097628179280, 12.74957792133969, 12.98813812741268
