| L(s) = 1 | − 2-s − 2.20·3-s + 4-s + 2.02·5-s + 2.20·6-s + 2.12·7-s − 8-s + 1.87·9-s − 2.02·10-s + 11-s − 2.20·12-s + 0.635·13-s − 2.12·14-s − 4.46·15-s + 16-s + 3.24·17-s − 1.87·18-s + 1.22·19-s + 2.02·20-s − 4.68·21-s − 22-s − 5.39·23-s + 2.20·24-s − 0.898·25-s − 0.635·26-s + 2.49·27-s + 2.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.27·3-s + 0.5·4-s + 0.905·5-s + 0.901·6-s + 0.802·7-s − 0.353·8-s + 0.623·9-s − 0.640·10-s + 0.301·11-s − 0.637·12-s + 0.176·13-s − 0.567·14-s − 1.15·15-s + 0.250·16-s + 0.785·17-s − 0.441·18-s + 0.281·19-s + 0.452·20-s − 1.02·21-s − 0.213·22-s − 1.12·23-s + 0.450·24-s − 0.179·25-s − 0.124·26-s + 0.479·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 13 | \( 1 - 0.635T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 + 0.808T + 41T^{2} \) |
| 43 | \( 1 + 4.72T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 + 9.29T + 71T^{2} \) |
| 73 | \( 1 + 7.54T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.984638506465281505013024063761, −7.28291120834046783351006330433, −6.42330294908654248637941049482, −5.76762491711483965000736271260, −5.41432425792162489105553251175, −4.45806384984886938250208718235, −3.29451255212643778705177338210, −1.92065829926548542985871499894, −1.36832165878406490913664355621, 0,
1.36832165878406490913664355621, 1.92065829926548542985871499894, 3.29451255212643778705177338210, 4.45806384984886938250208718235, 5.41432425792162489105553251175, 5.76762491711483965000736271260, 6.42330294908654248637941049482, 7.28291120834046783351006330433, 7.984638506465281505013024063761
