| L(s) = 1 | + 2.62·3-s − 1.76·5-s + 2.03·7-s + 3.87·9-s − 4.58·11-s − 0.991·13-s − 4.62·15-s − 5.18·17-s + 2.20·19-s + 5.33·21-s − 8.16·23-s − 1.88·25-s + 2.28·27-s + 0.471·29-s + 3.12·31-s − 12.0·33-s − 3.59·35-s − 4.70·37-s − 2.59·39-s + 4.39·41-s − 1.03·43-s − 6.83·45-s − 10.4·47-s − 2.85·49-s − 13.5·51-s + 8.99·53-s + 8.09·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.789·5-s + 0.769·7-s + 1.29·9-s − 1.38·11-s − 0.274·13-s − 1.19·15-s − 1.25·17-s + 0.506·19-s + 1.16·21-s − 1.70·23-s − 0.376·25-s + 0.439·27-s + 0.0875·29-s + 0.560·31-s − 2.09·33-s − 0.608·35-s − 0.772·37-s − 0.416·39-s + 0.685·41-s − 0.157·43-s − 1.01·45-s − 1.52·47-s − 0.407·49-s − 1.90·51-s + 1.23·53-s + 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 + 0.991T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 - 0.471T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 8.99T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 + 1.78T + 83T^{2} \) |
| 89 | \( 1 + 8.79T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.282006025616135325032611676724, −7.65135860623444998992948002238, −6.99817911171661251889078865244, −5.78900033820759678037311978856, −4.77868372671601212782926536361, −4.18675112760899871869354807238, −3.33292072188199103134748039215, −2.47436890642211265156447783260, −1.83657788765643404734963882799, 0,
1.83657788765643404734963882799, 2.47436890642211265156447783260, 3.33292072188199103134748039215, 4.18675112760899871869354807238, 4.77868372671601212782926536361, 5.78900033820759678037311978856, 6.99817911171661251889078865244, 7.65135860623444998992948002238, 8.282006025616135325032611676724
