| L(s) = 1 | + 2.24·2-s + 2.18·3-s + 3.04·4-s + 0.602·5-s + 4.90·6-s + 4.68·7-s + 2.33·8-s + 1.76·9-s + 1.35·10-s − 5.56·11-s + 6.63·12-s − 13-s + 10.5·14-s + 1.31·15-s − 0.832·16-s + 2.79·17-s + 3.96·18-s − 2.30·19-s + 1.83·20-s + 10.2·21-s − 12.4·22-s − 2.27·23-s + 5.10·24-s − 4.63·25-s − 2.24·26-s − 2.69·27-s + 14.2·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.26·3-s + 1.52·4-s + 0.269·5-s + 2.00·6-s + 1.76·7-s + 0.826·8-s + 0.588·9-s + 0.427·10-s − 1.67·11-s + 1.91·12-s − 0.277·13-s + 2.80·14-s + 0.339·15-s − 0.208·16-s + 0.677·17-s + 0.934·18-s − 0.529·19-s + 0.409·20-s + 2.22·21-s − 2.66·22-s − 0.473·23-s + 1.04·24-s − 0.927·25-s − 0.440·26-s − 0.518·27-s + 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.426966085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.426966085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 0.602T + 5T^{2} \) |
| 7 | \( 1 - 4.68T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + 0.322T + 29T^{2} \) |
| 31 | \( 1 + 9.68T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 0.717T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 + 9.44T + 73T^{2} \) |
| 79 | \( 1 + 0.560T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 8.91T + 97T^{2} \) |
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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
−9.585201225083756886082183040031, −8.551554862997752273190723697122, −7.79166184056090693454183422603, −7.45196366749460778240688949948, −5.80169979138435414703580405956, −5.32670578791035396408080667498, −4.45369030904166023829328637822, −3.61681905463701640401203612446, −2.43170325029180486961138128723, −2.07085753525597495228396200417,
2.07085753525597495228396200417, 2.43170325029180486961138128723, 3.61681905463701640401203612446, 4.45369030904166023829328637822, 5.32670578791035396408080667498, 5.80169979138435414703580405956, 7.45196366749460778240688949948, 7.79166184056090693454183422603, 8.551554862997752273190723697122, 9.585201225083756886082183040031
