| L(s) = 1 | − 18.3·2-s − 81·3-s − 173.·4-s + 1.62e3·5-s + 1.48e3·6-s + 7.42e3·7-s + 1.26e4·8-s + 6.56e3·9-s − 2.98e4·10-s − 7.47e4·11-s + 1.40e4·12-s − 3.15e4·13-s − 1.36e5·14-s − 1.31e5·15-s − 1.43e5·16-s + 2.75e5·17-s − 1.20e5·18-s + 1.30e5·19-s − 2.82e5·20-s − 6.01e5·21-s + 1.37e6·22-s − 1.03e6·23-s − 1.02e6·24-s + 6.86e5·25-s + 5.79e5·26-s − 5.31e5·27-s − 1.29e6·28-s + ⋯ |
| L(s) = 1 | − 0.812·2-s − 0.577·3-s − 0.339·4-s + 1.16·5-s + 0.469·6-s + 1.16·7-s + 1.08·8-s + 0.333·9-s − 0.944·10-s − 1.53·11-s + 0.195·12-s − 0.306·13-s − 0.950·14-s − 0.671·15-s − 0.545·16-s + 0.799·17-s − 0.270·18-s + 0.229·19-s − 0.394·20-s − 0.674·21-s + 1.25·22-s − 0.771·23-s − 0.628·24-s + 0.351·25-s + 0.248·26-s − 0.192·27-s − 0.396·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.176933595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176933595\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 + 18.3T + 512T^{2} \) |
| 5 | \( 1 - 1.62e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.42e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.47e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.15e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.75e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 1.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.07e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.42e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.67e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.58e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.46e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.15e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.67e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.39e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.55e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.80e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.85e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.79e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.28e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.23e8T + 7.60e17T^{2} \) |
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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
−13.45098227953455890143237215494, −12.01604450528439635890335197297, −10.40071962435550651390092212582, −10.11742513034691025141163551454, −8.536813422571917357286852728546, −7.53254064425732765038100810834, −5.62243600178029516227108758436, −4.78411227514485472029163337560, −2.12959690721544624881011367216, −0.826245364016455317476660575095,
0.826245364016455317476660575095, 2.12959690721544624881011367216, 4.78411227514485472029163337560, 5.62243600178029516227108758436, 7.53254064425732765038100810834, 8.536813422571917357286852728546, 10.11742513034691025141163551454, 10.40071962435550651390092212582, 12.01604450528439635890335197297, 13.45098227953455890143237215494
