| L(s) = 1 | + 3-s + 1.37·5-s + 7-s + 9-s − 4·11-s + 1.37·13-s + 1.37·15-s − 4.74·17-s − 4·19-s + 21-s + 23-s − 3.11·25-s + 27-s + 9.37·29-s − 6.74·31-s − 4·33-s + 1.37·35-s + 2.62·37-s + 1.37·39-s − 8.11·41-s − 6.11·43-s + 1.37·45-s − 4.62·47-s + 49-s − 4.74·51-s + 4.74·53-s − 5.48·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.613·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s + 0.380·13-s + 0.354·15-s − 1.15·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.623·25-s + 0.192·27-s + 1.74·29-s − 1.21·31-s − 0.696·33-s + 0.231·35-s + 0.431·37-s + 0.219·39-s − 1.26·41-s − 0.932·43-s + 0.204·45-s − 0.675·47-s + 0.142·49-s − 0.664·51-s + 0.651·53-s − 0.740·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−7.61281398134608590732092216128, −6.79838029264534286585467225560, −6.22024323730082386104415434298, −5.32637321486472345150759045410, −4.72339418726495577628536825309, −3.94119327514708601803431821617, −2.92626084659497970937035985799, −2.26307948419910956403881145132, −1.54920649876814086868171167078, 0,
1.54920649876814086868171167078, 2.26307948419910956403881145132, 2.92626084659497970937035985799, 3.94119327514708601803431821617, 4.72339418726495577628536825309, 5.32637321486472345150759045410, 6.22024323730082386104415434298, 6.79838029264534286585467225560, 7.61281398134608590732092216128
