| L(s) = 1 | − 2.18·3-s − 1.35·5-s + 2.12·7-s + 1.76·9-s + 1.59·11-s − 2.15·13-s + 2.96·15-s − 17-s − 2.52·19-s − 4.63·21-s + 7.50·23-s − 3.15·25-s + 2.69·27-s − 0.561·29-s + 2.91·31-s − 3.48·33-s − 2.88·35-s + 2.69·37-s + 4.71·39-s − 0.852·41-s − 8.77·43-s − 2.39·45-s + 5.59·47-s − 2.49·49-s + 2.18·51-s + 5.14·53-s − 2.16·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.607·5-s + 0.802·7-s + 0.588·9-s + 0.481·11-s − 0.598·13-s + 0.765·15-s − 0.242·17-s − 0.578·19-s − 1.01·21-s + 1.56·23-s − 0.631·25-s + 0.518·27-s − 0.104·29-s + 0.523·31-s − 0.606·33-s − 0.487·35-s + 0.443·37-s + 0.754·39-s − 0.133·41-s − 1.33·43-s − 0.357·45-s + 0.816·47-s − 0.356·49-s + 0.305·51-s + 0.707·53-s − 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9239514295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9239514295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 + 0.852T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 61 | \( 1 - 0.537T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 0.578T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 0.663T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 0.927T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−7.73083693354475921824885385716, −7.03009845390032795504063854407, −6.47409954578585746487381532701, −5.70556134140204963472029642219, −4.92697353604033114602390661775, −4.59392731899703358584281355525, −3.70794306136714802295085016285, −2.63410331196608198096392572291, −1.52010744037899930198664639226, −0.53103995026899488506252910805,
0.53103995026899488506252910805, 1.52010744037899930198664639226, 2.63410331196608198096392572291, 3.70794306136714802295085016285, 4.59392731899703358584281355525, 4.92697353604033114602390661775, 5.70556134140204963472029642219, 6.47409954578585746487381532701, 7.03009845390032795504063854407, 7.73083693354475921824885385716
