| L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s + 0.529·5-s + 0.470·6-s + 7-s − 1.77·8-s + 9-s + 0.249·10-s + 4.77·11-s − 1.77·12-s + 5.02·13-s + 0.470·14-s + 0.529·15-s + 2.71·16-s + 17-s + 0.470·18-s − 1.47·19-s − 0.941·20-s + 21-s + 2.24·22-s − 0.778·23-s − 1.77·24-s − 4.71·25-s + 2.36·26-s + 27-s − 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s + 0.236·5-s + 0.192·6-s + 0.377·7-s − 0.628·8-s + 0.333·9-s + 0.0787·10-s + 1.44·11-s − 0.513·12-s + 1.39·13-s + 0.125·14-s + 0.136·15-s + 0.679·16-s + 0.242·17-s + 0.110·18-s − 0.337·19-s − 0.210·20-s + 0.218·21-s + 0.479·22-s − 0.162·23-s − 0.363·24-s − 0.943·25-s + 0.464·26-s + 0.192·27-s − 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.788294337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788294337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 0.778T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 8.86T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 3.30T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 8.05T + 89T^{2} \) |
| 97 | \( 1 - 3.55T + 97T^{2} \) |
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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
−11.59485497294390259704713401527, −10.44978117422502264167856176870, −9.311964032565132756369545544178, −8.845872608504172513453441260467, −7.925336326812937879737117327477, −6.51658074289486414126472186181, −5.54158756840717382234083783981, −4.16104958825721235092395451518, −3.54178766549477272067615419805, −1.54369465786752064082182784273,
1.54369465786752064082182784273, 3.54178766549477272067615419805, 4.16104958825721235092395451518, 5.54158756840717382234083783981, 6.51658074289486414126472186181, 7.925336326812937879737117327477, 8.845872608504172513453441260467, 9.311964032565132756369545544178, 10.44978117422502264167856176870, 11.59485497294390259704713401527
