| L(s) = 1 | − 0.918·3-s + 5-s − 0.478·7-s − 2.15·9-s − 4.45·11-s + 3.37·13-s − 0.918·15-s − 7.40·17-s − 2.45·19-s + 0.439·21-s + 7.00·23-s + 25-s + 4.73·27-s + 29-s + 7.65·31-s + 4.08·33-s − 0.478·35-s − 7.79·37-s − 3.09·39-s + 7.12·41-s + 11.9·43-s − 2.15·45-s + 3.24·47-s − 6.77·49-s + 6.80·51-s + 13.2·53-s − 4.45·55-s + ⋯ |
| L(s) = 1 | − 0.530·3-s + 0.447·5-s − 0.180·7-s − 0.718·9-s − 1.34·11-s + 0.935·13-s − 0.237·15-s − 1.79·17-s − 0.562·19-s + 0.0959·21-s + 1.46·23-s + 0.200·25-s + 0.911·27-s + 0.185·29-s + 1.37·31-s + 0.711·33-s − 0.0809·35-s − 1.28·37-s − 0.495·39-s + 1.11·41-s + 1.82·43-s − 0.321·45-s + 0.472·47-s − 0.967·49-s + 0.952·51-s + 1.81·53-s − 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.918T + 3T^{2} \) |
| 7 | \( 1 + 0.478T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 7.40T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 7.00T + 23T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 - 9.37T + 67T^{2} \) |
| 71 | \( 1 + 6.78T + 71T^{2} \) |
| 73 | \( 1 + 9.38T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−7.22164583224495372366056723537, −6.54783085116247302350429747291, −6.03549087781648645031780682320, −5.34640104208500885279459113092, −4.75721951154435984168413023338, −3.93240333146922059926289263335, −2.67947545681482301698345437623, −2.50804378872110469419386242700, −1.06190947338254059152862627628, 0,
1.06190947338254059152862627628, 2.50804378872110469419386242700, 2.67947545681482301698345437623, 3.93240333146922059926289263335, 4.75721951154435984168413023338, 5.34640104208500885279459113092, 6.03549087781648645031780682320, 6.54783085116247302350429747291, 7.22164583224495372366056723537
