L(s) = 1 | − 2.90·3-s − 3.52·7-s + 5.42·9-s − 3.80·11-s − 2.62·13-s + 5.80·17-s + 5.05·19-s + 10.2·21-s − 0.474·23-s − 7.05·27-s + 2·29-s + 2.75·31-s + 11.0·33-s − 7.18·37-s + 7.61·39-s + 5.18·41-s + 1.95·43-s + 5.33·47-s + 5.42·49-s − 16.8·51-s − 5.37·53-s − 14.6·57-s − 5.05·59-s + 12.2·61-s − 19.1·63-s + 7.76·67-s + 1.37·69-s + ⋯ |
L(s) = 1 | − 1.67·3-s − 1.33·7-s + 1.80·9-s − 1.14·11-s − 0.727·13-s + 1.40·17-s + 1.15·19-s + 2.23·21-s − 0.0989·23-s − 1.35·27-s + 0.371·29-s + 0.494·31-s + 1.92·33-s − 1.18·37-s + 1.21·39-s + 0.809·41-s + 0.297·43-s + 0.777·47-s + 0.775·49-s − 2.36·51-s − 0.738·53-s − 1.94·57-s − 0.657·59-s + 1.56·61-s − 2.41·63-s + 0.948·67-s + 0.165·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 0.474T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.95T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 5.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.061916963971535515451910583893, −7.28848271779358453382459274823, −6.75599148957352270341634295761, −5.78490112392852066806254395019, −5.47869448250385351962819572016, −4.71366110708683662573541108506, −3.53213132492645684856537405150, −2.66639596194841814838482813479, −1.00155140567354812934813386066, 0,
1.00155140567354812934813386066, 2.66639596194841814838482813479, 3.53213132492645684856537405150, 4.71366110708683662573541108506, 5.47869448250385351962819572016, 5.78490112392852066806254395019, 6.75599148957352270341634295761, 7.28848271779358453382459274823, 8.061916963971535515451910583893