| L(s) = 1 | + 19.0·2-s + 219.·3-s − 147.·4-s + 1.17e3·5-s + 4.19e3·6-s + 4.84e3·7-s − 1.25e4·8-s + 2.86e4·9-s + 2.23e4·10-s + 2.02e4·11-s − 3.23e4·12-s + 4.37e4·13-s + 9.25e4·14-s + 2.57e5·15-s − 1.64e5·16-s + 2.67e5·17-s + 5.47e5·18-s + 1.10e5·19-s − 1.72e5·20-s + 1.06e6·21-s + 3.86e5·22-s − 1.86e6·23-s − 2.76e6·24-s − 5.80e5·25-s + 8.35e5·26-s + 1.97e6·27-s − 7.14e5·28-s + ⋯ |
| L(s) = 1 | + 0.843·2-s + 1.56·3-s − 0.287·4-s + 0.838·5-s + 1.32·6-s + 0.762·7-s − 1.08·8-s + 1.45·9-s + 0.707·10-s + 0.416·11-s − 0.450·12-s + 0.424·13-s + 0.643·14-s + 1.31·15-s − 0.629·16-s + 0.778·17-s + 1.22·18-s + 0.195·19-s − 0.241·20-s + 1.19·21-s + 0.351·22-s − 1.38·23-s − 1.70·24-s − 0.297·25-s + 0.358·26-s + 0.713·27-s − 0.219·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.272492268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.272492268\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 19.0T + 512T^{2} \) |
| 3 | \( 1 - 219.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.17e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.84e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.02e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.37e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.67e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.10e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.86e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.07e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.62e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.06e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 3.05e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.63e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.81e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.73e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.99e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.54e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.11e9T + 7.60e17T^{2} \) |
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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
−14.14439640391404591933339279450, −13.33998443207835825999193167111, −12.03546117425446454082252022716, −9.989744671526382706337164003882, −8.972887776426503393366592947622, −7.945759637277146911343434062160, −5.94605606118501308275318381628, −4.37000111747986949037402193079, −3.10737231380697863265776721411, −1.67899736806803374392737723873,
1.67899736806803374392737723873, 3.10737231380697863265776721411, 4.37000111747986949037402193079, 5.94605606118501308275318381628, 7.945759637277146911343434062160, 8.972887776426503393366592947622, 9.989744671526382706337164003882, 12.03546117425446454082252022716, 13.33998443207835825999193167111, 14.14439640391404591933339279450
