| L(s) = 1 | + 2-s − 0.366·3-s + 4-s + 1.52·5-s − 0.366·6-s − 3.90·7-s + 8-s − 2.86·9-s + 1.52·10-s − 4.33·11-s − 0.366·12-s + 4.63·13-s − 3.90·14-s − 0.557·15-s + 16-s − 4.69·17-s − 2.86·18-s − 2.13·19-s + 1.52·20-s + 1.43·21-s − 4.33·22-s − 23-s − 0.366·24-s − 2.68·25-s + 4.63·26-s + 2.14·27-s − 3.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.211·3-s + 0.5·4-s + 0.680·5-s − 0.149·6-s − 1.47·7-s + 0.353·8-s − 0.955·9-s + 0.481·10-s − 1.30·11-s − 0.105·12-s + 1.28·13-s − 1.04·14-s − 0.144·15-s + 0.250·16-s − 1.13·17-s − 0.675·18-s − 0.488·19-s + 0.340·20-s + 0.312·21-s − 0.923·22-s − 0.208·23-s − 0.0747·24-s − 0.536·25-s + 0.908·26-s + 0.413·27-s − 0.737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.366T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 4.63T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 + 2.13T + 19T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 0.963T + 47T^{2} \) |
| 53 | \( 1 + 0.987T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 + 0.321T + 79T^{2} \) |
| 83 | \( 1 + 7.83T + 83T^{2} \) |
| 89 | \( 1 + 6.86T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.166113789542353337104967162168, −8.560683159180055906762994902850, −7.35443905331050139994061020311, −6.31142456408225805334935769326, −5.99094986316871185351729598980, −5.21247065997826531840338440149, −3.90460108703822978764532297543, −3.02062748226055301919793471534, −2.14734943206522085701623544266, 0,
2.14734943206522085701623544266, 3.02062748226055301919793471534, 3.90460108703822978764532297543, 5.21247065997826531840338440149, 5.99094986316871185351729598980, 6.31142456408225805334935769326, 7.35443905331050139994061020311, 8.560683159180055906762994902850, 9.166113789542353337104967162168
