L(s) = 1 | + 2.66·3-s − 5-s + 1.97·7-s + 4.08·9-s − 1.73·11-s − 1.28·13-s − 2.66·15-s − 5.75·17-s + 5.34·19-s + 5.25·21-s − 8.91·23-s + 25-s + 2.88·27-s − 9.63·29-s − 2.06·31-s − 4.62·33-s − 1.97·35-s − 1.44·37-s − 3.43·39-s + 1.37·41-s + 2.31·43-s − 4.08·45-s − 6.46·47-s − 3.10·49-s − 15.3·51-s − 2.84·53-s + 1.73·55-s + ⋯ |
L(s) = 1 | + 1.53·3-s − 0.447·5-s + 0.745·7-s + 1.36·9-s − 0.523·11-s − 0.357·13-s − 0.687·15-s − 1.39·17-s + 1.22·19-s + 1.14·21-s − 1.85·23-s + 0.200·25-s + 0.556·27-s − 1.78·29-s − 0.371·31-s − 0.804·33-s − 0.333·35-s − 0.238·37-s − 0.549·39-s + 0.214·41-s + 0.352·43-s − 0.609·45-s − 0.943·47-s − 0.443·49-s − 2.14·51-s − 0.391·53-s + 0.234·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 8.91T + 23T^{2} \) |
| 29 | \( 1 + 9.63T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + 0.511T + 59T^{2} \) |
| 61 | \( 1 + 0.496T + 61T^{2} \) |
| 67 | \( 1 - 0.713T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 - 3.45T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.67425077697401415130994680467, −7.23557878264059265782611954507, −6.14964156879394283171569304745, −5.23148628467560247183750679026, −4.46024737720731844485214691239, −3.80920597012635408840382036945, −3.10714914498094898357431973674, −2.18840371491872878602317483153, −1.71479755427052812727900853673, 0,
1.71479755427052812727900853673, 2.18840371491872878602317483153, 3.10714914498094898357431973674, 3.80920597012635408840382036945, 4.46024737720731844485214691239, 5.23148628467560247183750679026, 6.14964156879394283171569304745, 7.23557878264059265782611954507, 7.67425077697401415130994680467