| L(s) = 1 | + 0.585·3-s − 5-s − 2·7-s − 2.65·9-s + 4.24·11-s + 13-s − 0.585·15-s + 0.828·17-s − 0.242·19-s − 1.17·21-s − 9.07·23-s + 25-s − 3.31·27-s − 1.65·29-s + 1.41·31-s + 2.48·33-s + 2·35-s + 6.82·37-s + 0.585·39-s + 4.82·41-s + 10.2·43-s + 2.65·45-s + 2·47-s − 3·49-s + 0.485·51-s + 8.82·53-s − 4.24·55-s + ⋯ |
| L(s) = 1 | + 0.338·3-s − 0.447·5-s − 0.755·7-s − 0.885·9-s + 1.27·11-s + 0.277·13-s − 0.151·15-s + 0.200·17-s − 0.0556·19-s − 0.255·21-s − 1.89·23-s + 0.200·25-s − 0.637·27-s − 0.307·29-s + 0.254·31-s + 0.432·33-s + 0.338·35-s + 1.12·37-s + 0.0938·39-s + 0.754·41-s + 1.56·43-s + 0.396·45-s + 0.291·47-s − 0.428·49-s + 0.0679·51-s + 1.21·53-s − 0.572·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552413042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552413042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.585T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 0.242T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 2.58T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 4.82T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.328441940426571056855711683931, −7.84970050872286993144302044287, −6.93357160234721462092719288322, −6.09583427748981082065820691853, −5.77216933953187666372579590041, −4.33177534068563219174435031995, −3.83650278859712119376000546895, −3.03905955994187989353330583639, −2.07893166877489726131678924104, −0.68205701631050462587658961406,
0.68205701631050462587658961406, 2.07893166877489726131678924104, 3.03905955994187989353330583639, 3.83650278859712119376000546895, 4.33177534068563219174435031995, 5.77216933953187666372579590041, 6.09583427748981082065820691853, 6.93357160234721462092719288322, 7.84970050872286993144302044287, 8.328441940426571056855711683931
