| L(s) = 1 | + 16·2-s − 206.·3-s + 256·4-s − 845.·5-s − 3.30e3·6-s − 6.60e3·7-s + 4.09e3·8-s + 2.31e4·9-s − 1.35e4·10-s + 5.64e4·11-s − 5.29e4·12-s − 1.01e5·13-s − 1.05e5·14-s + 1.74e5·15-s + 6.55e4·16-s + 6.24e5·17-s + 3.69e5·18-s − 1.30e5·19-s − 2.16e5·20-s + 1.36e6·21-s + 9.03e5·22-s + 2.61e6·23-s − 8.47e5·24-s − 1.23e6·25-s − 1.62e6·26-s − 7.07e5·27-s − 1.69e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s − 0.604·5-s − 1.04·6-s − 1.04·7-s + 0.353·8-s + 1.17·9-s − 0.427·10-s + 1.16·11-s − 0.737·12-s − 0.984·13-s − 0.735·14-s + 0.891·15-s + 0.250·16-s + 1.81·17-s + 0.829·18-s − 0.229·19-s − 0.302·20-s + 1.53·21-s + 0.822·22-s + 1.95·23-s − 0.521·24-s − 0.634·25-s − 0.696·26-s − 0.256·27-s − 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.345000625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345000625\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 3 | \( 1 + 206.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 845.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.60e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.64e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.01e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.24e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 2.61e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.86e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.44e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.66e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.70e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.25e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.34e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.24e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.70e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.67e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.72e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 9.46e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.50e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.03e7T + 7.60e17T^{2} \) |
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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
−14.33505573346858404760176349756, −12.54522393294348365180111839167, −12.14190031003350546835910069844, −11.00734793610807216250797245197, −9.663900724882127217429430551842, −7.24892077966604689948191128989, −6.21913989977530491476035016192, −4.96771480734055571258555767301, −3.43533471742143095075850093057, −0.78405756366905876824752866698,
0.78405756366905876824752866698, 3.43533471742143095075850093057, 4.96771480734055571258555767301, 6.21913989977530491476035016192, 7.24892077966604689948191128989, 9.663900724882127217429430551842, 11.00734793610807216250797245197, 12.14190031003350546835910069844, 12.54522393294348365180111839167, 14.33505573346858404760176349756
