L(s) = 1 | − 3-s + 2.45·5-s − 2.43·7-s + 9-s − 2.22·11-s − 1.05·13-s − 2.45·15-s + 0.227·17-s + 2.91·19-s + 2.43·21-s − 2.55·23-s + 1.02·25-s − 27-s − 0.104·29-s + 3.77·31-s + 2.22·33-s − 5.98·35-s − 0.875·37-s + 1.05·39-s + 7.02·41-s + 1.10·43-s + 2.45·45-s − 0.498·47-s − 1.06·49-s − 0.227·51-s + 5.62·53-s − 5.47·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.09·5-s − 0.920·7-s + 0.333·9-s − 0.672·11-s − 0.293·13-s − 0.633·15-s + 0.0551·17-s + 0.669·19-s + 0.531·21-s − 0.532·23-s + 0.205·25-s − 0.192·27-s − 0.0193·29-s + 0.677·31-s + 0.388·33-s − 1.01·35-s − 0.143·37-s + 0.169·39-s + 1.09·41-s + 0.168·43-s + 0.366·45-s − 0.0727·47-s − 0.152·49-s − 0.0318·51-s + 0.772·53-s − 0.738·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 0.227T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 + 0.104T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 + 0.875T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 + 0.498T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 + 9.29T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 1.00T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−7.37409035679675306193439287977, −6.65338415713537302157083007001, −5.99171487161207508482198131229, −5.59315964067262762913075402650, −4.85324563062329663255589563420, −3.94272305776270274935276560549, −2.93787055977163865361775028138, −2.29778650659006345349623445344, −1.20917530742033982658048586284, 0,
1.20917530742033982658048586284, 2.29778650659006345349623445344, 2.93787055977163865361775028138, 3.94272305776270274935276560549, 4.85324563062329663255589563420, 5.59315964067262762913075402650, 5.99171487161207508482198131229, 6.65338415713537302157083007001, 7.37409035679675306193439287977