| L(s) = 1 | − 0.745·3-s − 0.935·5-s − 7-s − 2.44·9-s − 0.508·11-s + 5.36·13-s + 0.697·15-s − 17-s − 5.87·19-s + 0.745·21-s − 3.36·23-s − 4.12·25-s + 4.06·27-s − 1.87·29-s − 3.91·31-s + 0.379·33-s + 0.935·35-s + 2.12·37-s − 4·39-s − 9.12·41-s − 2.10·43-s + 2.28·45-s + 2.98·47-s + 49-s + 0.745·51-s + 11.3·53-s + 0.475·55-s + ⋯ |
| L(s) = 1 | − 0.430·3-s − 0.418·5-s − 0.377·7-s − 0.814·9-s − 0.153·11-s + 1.48·13-s + 0.180·15-s − 0.242·17-s − 1.34·19-s + 0.162·21-s − 0.701·23-s − 0.824·25-s + 0.781·27-s − 0.347·29-s − 0.703·31-s + 0.0659·33-s + 0.158·35-s + 0.350·37-s − 0.640·39-s − 1.42·41-s − 0.321·43-s + 0.340·45-s + 0.435·47-s + 0.142·49-s + 0.104·51-s + 1.55·53-s + 0.0641·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7732188619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7732188619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.745T + 3T^{2} \) |
| 5 | \( 1 + 0.935T + 5T^{2} \) |
| 11 | \( 1 + 0.508T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 6.14T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 9.28T + 97T^{2} \) |
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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
−8.077365073837478323423977594673, −7.02333800273168016789032966009, −6.42677664200808133019595086414, −5.81433488136102177958514243240, −5.27088768600363420127331681693, −4.02362040332324347804744828855, −3.80492250942555462135149779891, −2.70987575125322459964159157891, −1.77507553752995835058949062114, −0.43613490399659705165179248398,
0.43613490399659705165179248398, 1.77507553752995835058949062114, 2.70987575125322459964159157891, 3.80492250942555462135149779891, 4.02362040332324347804744828855, 5.27088768600363420127331681693, 5.81433488136102177958514243240, 6.42677664200808133019595086414, 7.02333800273168016789032966009, 8.077365073837478323423977594673
