| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2.24·11-s − 5.41·13-s + 15-s − 4.41·17-s − 7.07·19-s − 21-s − 23-s + 25-s + 27-s + 1.24·29-s − 10.6·31-s + 2.24·33-s − 35-s − 3·37-s − 5.41·39-s − 1.58·41-s + 2·43-s + 45-s + 5.41·47-s − 6·49-s − 4.41·51-s − 6.41·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.676·11-s − 1.50·13-s + 0.258·15-s − 1.07·17-s − 1.62·19-s − 0.218·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.230·29-s − 1.91·31-s + 0.390·33-s − 0.169·35-s − 0.493·37-s − 0.866·39-s − 0.247·41-s + 0.304·43-s + 0.149·45-s + 0.789·47-s − 0.857·49-s − 0.618·51-s − 0.881·53-s + 0.302·55-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 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 2.07T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−8.557881248032575575031632750896, −7.69572262104127094017501188922, −6.80439439741677106487913133266, −6.40418351075388646092732051039, −5.23706384510496627759867174103, −4.42346385976418431518706366127, −3.60341359730816206039508221907, −2.44098287001182138375579430883, −1.87057519701961739989367761995, 0,
1.87057519701961739989367761995, 2.44098287001182138375579430883, 3.60341359730816206039508221907, 4.42346385976418431518706366127, 5.23706384510496627759867174103, 6.40418351075388646092732051039, 6.80439439741677106487913133266, 7.69572262104127094017501188922, 8.557881248032575575031632750896
