| L(s) = 1 | + 14.3·2-s − 81·3-s − 306.·4-s − 1.16e3·6-s + 2.87e3·7-s − 1.17e4·8-s + 6.56e3·9-s − 2.32e4·11-s + 2.48e4·12-s − 1.12e5·13-s + 4.12e4·14-s − 1.13e4·16-s − 1.15e5·17-s + 9.40e4·18-s − 2.13e5·19-s − 2.33e5·21-s − 3.33e5·22-s + 1.83e6·23-s + 9.50e5·24-s − 1.61e6·26-s − 5.31e5·27-s − 8.82e5·28-s + 3.67e6·29-s + 8.85e6·31-s + 5.84e6·32-s + 1.88e6·33-s − 1.66e6·34-s + ⋯ |
| L(s) = 1 | + 0.633·2-s − 0.577·3-s − 0.598·4-s − 0.365·6-s + 0.453·7-s − 1.01·8-s + 0.333·9-s − 0.479·11-s + 0.345·12-s − 1.09·13-s + 0.287·14-s − 0.0432·16-s − 0.336·17-s + 0.211·18-s − 0.375·19-s − 0.261·21-s − 0.303·22-s + 1.36·23-s + 0.584·24-s − 0.692·26-s − 0.192·27-s − 0.271·28-s + 0.964·29-s + 1.72·31-s + 0.985·32-s + 0.276·33-s − 0.213·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.527604735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527604735\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 14.3T + 512T^{2} \) |
| 7 | \( 1 - 2.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.12e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.15e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.13e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.83e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.17e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.93e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.26e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.02e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.56e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.38e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.40e9T + 7.60e17T^{2} \) |
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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
−12.68410524549980840602236762680, −11.82069001605622367688101716488, −10.55529936272009203939657296150, −9.360243459629158087547393579834, −8.030403381197734361316903164932, −6.52275362174859700813558300115, −5.14093999178245151214751952365, −4.48447872206564397506837389961, −2.72385558781020656100769999039, −0.68472310797918910595708570596,
0.68472310797918910595708570596, 2.72385558781020656100769999039, 4.48447872206564397506837389961, 5.14093999178245151214751952365, 6.52275362174859700813558300115, 8.030403381197734361316903164932, 9.360243459629158087547393579834, 10.55529936272009203939657296150, 11.82069001605622367688101716488, 12.68410524549980840602236762680
