| L(s) = 1 | + 1.54·2-s + 2.24·3-s + 0.375·4-s − 5-s + 3.45·6-s + 3.95·7-s − 2.50·8-s + 2.02·9-s − 1.54·10-s + 1.30·11-s + 0.841·12-s + 4.16·13-s + 6.10·14-s − 2.24·15-s − 4.60·16-s − 5.17·17-s + 3.12·18-s + 5.17·19-s − 0.375·20-s + 8.87·21-s + 2.00·22-s + 2.34·23-s − 5.61·24-s + 25-s + 6.41·26-s − 2.17·27-s + 1.48·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.29·3-s + 0.187·4-s − 0.447·5-s + 1.41·6-s + 1.49·7-s − 0.885·8-s + 0.676·9-s − 0.487·10-s + 0.393·11-s + 0.242·12-s + 1.15·13-s + 1.63·14-s − 0.579·15-s − 1.15·16-s − 1.25·17-s + 0.737·18-s + 1.18·19-s − 0.0838·20-s + 1.93·21-s + 0.428·22-s + 0.488·23-s − 1.14·24-s + 0.200·25-s + 1.25·26-s − 0.419·27-s + 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.291389136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.291389136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 0.379T + 31T^{2} \) |
| 37 | \( 1 + 0.812T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 + 0.246T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 1.04T + 73T^{2} \) |
| 79 | \( 1 - 9.26T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−9.423759827632691401229590932972, −8.633873933002279040341860902019, −8.377510660622146477065457071648, −7.36469249053573985714190218269, −6.32291147199648061912023906637, −5.13660615110900555363619705078, −4.44965804753448070659673410994, −3.64177851414149121604225835404, −2.83612323929727641939085759103, −1.54489318814831868412917613868,
1.54489318814831868412917613868, 2.83612323929727641939085759103, 3.64177851414149121604225835404, 4.44965804753448070659673410994, 5.13660615110900555363619705078, 6.32291147199648061912023906637, 7.36469249053573985714190218269, 8.377510660622146477065457071648, 8.633873933002279040341860902019, 9.423759827632691401229590932972
