| L(s) = 1 | − 3.35·3-s − 9.35·5-s − 7·7-s − 15.7·9-s + 11·11-s − 35.5·13-s + 31.3·15-s − 67.4·17-s − 58.9·19-s + 23.4·21-s − 30.2·23-s − 37.5·25-s + 143.·27-s − 279.·29-s + 86.0·31-s − 36.8·33-s + 65.4·35-s − 38.7·37-s + 119.·39-s − 337.·41-s − 49.1·43-s + 147.·45-s − 223.·47-s + 49·49-s + 225.·51-s − 286.·53-s − 102.·55-s + ⋯ |
| L(s) = 1 | − 0.645·3-s − 0.836·5-s − 0.377·7-s − 0.583·9-s + 0.301·11-s − 0.757·13-s + 0.539·15-s − 0.961·17-s − 0.711·19-s + 0.243·21-s − 0.274·23-s − 0.300·25-s + 1.02·27-s − 1.79·29-s + 0.498·31-s − 0.194·33-s + 0.316·35-s − 0.172·37-s + 0.488·39-s − 1.28·41-s − 0.174·43-s + 0.488·45-s − 0.693·47-s + 0.142·49-s + 0.620·51-s − 0.743·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2004796174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2004796174\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 3.35T + 27T^{2} \) |
| 5 | \( 1 + 9.35T + 125T^{2} \) |
| 13 | \( 1 + 35.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 279.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 86.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 49.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 286.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 741.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 744.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 663.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 74.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 740.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 704.T + 9.12e5T^{2} \) |
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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
−9.303709212423638394413241788135, −8.522901896881293512684781113502, −7.67098828971359044050986776727, −6.77992282774079127536142026125, −6.07657734248593440532776979250, −5.07204832740554947810576911889, −4.20730622993960559534615824799, −3.24221831011902974824518140016, −1.99498523288508572864207090016, −0.22200957372730158737587154282,
0.22200957372730158737587154282, 1.99498523288508572864207090016, 3.24221831011902974824518140016, 4.20730622993960559534615824799, 5.07204832740554947810576911889, 6.07657734248593440532776979250, 6.77992282774079127536142026125, 7.67098828971359044050986776727, 8.522901896881293512684781113502, 9.303709212423638394413241788135
