| L(s) = 1 | + 2.25·3-s − 1.00·5-s + 0.939·7-s + 2.07·9-s − 1.56·11-s + 2.80·13-s − 2.25·15-s − 2.32·17-s − 1.86·19-s + 2.11·21-s + 0.0629·23-s − 3.99·25-s − 2.07·27-s − 8.13·31-s − 3.51·33-s − 0.940·35-s − 9.87·37-s + 6.32·39-s − 0.928·41-s + 5.51·43-s − 2.07·45-s − 4.88·47-s − 6.11·49-s − 5.23·51-s + 0.420·53-s + 1.56·55-s − 4.20·57-s + ⋯ |
| L(s) = 1 | + 1.30·3-s − 0.447·5-s + 0.355·7-s + 0.692·9-s − 0.470·11-s + 0.778·13-s − 0.582·15-s − 0.563·17-s − 0.427·19-s + 0.462·21-s + 0.0131·23-s − 0.799·25-s − 0.400·27-s − 1.46·31-s − 0.612·33-s − 0.158·35-s − 1.62·37-s + 1.01·39-s − 0.144·41-s + 0.840·43-s − 0.309·45-s − 0.712·47-s − 0.873·49-s − 0.733·51-s + 0.0578·53-s + 0.210·55-s − 0.556·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 - 0.939T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 + 2.32T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 - 0.0629T + 23T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 + 0.928T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 - 0.420T + 53T^{2} \) |
| 59 | \( 1 + 9.07T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 8.84T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.78541703836685947555737356735, −7.20805256393256240399957240308, −6.30940315165172160272339417798, −5.45601441251088107501926921192, −4.58262830889600414587730557832, −3.73644853421709217077509504928, −3.31877790169873329474868996196, −2.26591564288039428286288467608, −1.64534405833810060974367071570, 0,
1.64534405833810060974367071570, 2.26591564288039428286288467608, 3.31877790169873329474868996196, 3.73644853421709217077509504928, 4.58262830889600414587730557832, 5.45601441251088107501926921192, 6.30940315165172160272339417798, 7.20805256393256240399957240308, 7.78541703836685947555737356735
