| L(s) = 1 | − 3-s − 5-s − 0.845·7-s + 9-s − 4.55·11-s + 5.23·13-s + 15-s + 4.72·17-s − 5.54·19-s + 0.845·21-s + 23-s + 25-s − 27-s + 7.06·29-s − 1.83·31-s + 4.55·33-s + 0.845·35-s − 9.93·37-s − 5.23·39-s + 5.71·41-s + 4.78·43-s − 45-s + 5.54·47-s − 6.28·49-s − 4.72·51-s − 14.4·53-s + 4.55·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.319·7-s + 0.333·9-s − 1.37·11-s + 1.45·13-s + 0.258·15-s + 1.14·17-s − 1.27·19-s + 0.184·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.31·29-s − 0.329·31-s + 0.793·33-s + 0.142·35-s − 1.63·37-s − 0.838·39-s + 0.892·41-s + 0.729·43-s − 0.149·45-s + 0.808·47-s − 0.897·49-s − 0.661·51-s − 1.99·53-s + 0.614·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 0.845T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 5.54T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 5.06T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 7.85T + 67T^{2} \) |
| 71 | \( 1 - 3.71T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 7.50T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + 2.33T + 97T^{2} \) |
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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
−8.292195427376925883400102021894, −7.80024856832439229866743684437, −6.82284643348360774117457958989, −6.10039100263060776877884737575, −5.41202698053180082276466166372, −4.53261424905023199767286099425, −3.61938037826117689607176968170, −2.75164082853610513562628053135, −1.32242365775724753679596108695, 0,
1.32242365775724753679596108695, 2.75164082853610513562628053135, 3.61938037826117689607176968170, 4.53261424905023199767286099425, 5.41202698053180082276466166372, 6.10039100263060776877884737575, 6.82284643348360774117457958989, 7.80024856832439229866743684437, 8.292195427376925883400102021894
