| L(s) = 1 | + 0.474·3-s + 3.03·7-s − 2.77·9-s − 2·11-s + 1.42·13-s − 1.86·17-s − 0.903·19-s + 1.44·21-s − 3.32·23-s − 2.74·27-s + 3.96·29-s + 6.43·31-s − 0.949·33-s − 3.82·37-s + 0.677·39-s − 1.83·41-s − 3.59·43-s − 4.79·47-s + 2.21·49-s − 0.883·51-s + 9.50·53-s − 0.428·57-s − 10.6·59-s + 14.2·61-s − 8.42·63-s − 10.6·67-s − 1.58·69-s + ⋯ |
| L(s) = 1 | + 0.274·3-s + 1.14·7-s − 0.924·9-s − 0.603·11-s + 0.395·13-s − 0.451·17-s − 0.207·19-s + 0.314·21-s − 0.694·23-s − 0.527·27-s + 0.735·29-s + 1.15·31-s − 0.165·33-s − 0.628·37-s + 0.108·39-s − 0.286·41-s − 0.548·43-s − 0.700·47-s + 0.316·49-s − 0.123·51-s + 1.30·53-s − 0.0568·57-s − 1.38·59-s + 1.82·61-s − 1.06·63-s − 1.30·67-s − 0.190·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 0.474T + 3T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 + 0.903T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 3.59T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 0.267T + 73T^{2} \) |
| 79 | \( 1 - 8.57T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 - 9.95T + 97T^{2} \) |
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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
−7.51673270994486075435138791622, −6.57165772296147583754809521676, −5.95414350986263432588259323563, −5.16783298290714358443164668851, −4.65527787786121048792607703334, −3.80605650080793675895559531423, −2.88828832031221836042959699582, −2.23186077981393237687771594456, −1.32602518256030225572842769630, 0,
1.32602518256030225572842769630, 2.23186077981393237687771594456, 2.88828832031221836042959699582, 3.80605650080793675895559531423, 4.65527787786121048792607703334, 5.16783298290714358443164668851, 5.95414350986263432588259323563, 6.57165772296147583754809521676, 7.51673270994486075435138791622
