| L(s) = 1 | + 21.0·2-s + 27·3-s + 315.·4-s + 568.·6-s + 923.·7-s + 3.93e3·8-s + 729·9-s − 3.26e3·11-s + 8.50e3·12-s − 9.97e3·13-s + 1.94e4·14-s + 4.25e4·16-s − 6.00e3·17-s + 1.53e4·18-s + 3.43e4·19-s + 2.49e4·21-s − 6.87e4·22-s − 1.48e4·23-s + 1.06e5·24-s − 2.09e5·26-s + 1.96e4·27-s + 2.91e5·28-s − 8.41e4·29-s + 1.81e5·31-s + 3.91e5·32-s − 8.81e4·33-s − 1.26e5·34-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.577·3-s + 2.46·4-s + 1.07·6-s + 1.01·7-s + 2.71·8-s + 0.333·9-s − 0.739·11-s + 1.42·12-s − 1.25·13-s + 1.89·14-s + 2.59·16-s − 0.296·17-s + 0.620·18-s + 1.14·19-s + 0.587·21-s − 1.37·22-s − 0.255·23-s + 1.56·24-s − 2.34·26-s + 0.192·27-s + 2.50·28-s − 0.640·29-s + 1.09·31-s + 2.11·32-s − 0.427·33-s − 0.551·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.530823812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.530823812\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 21.0T + 128T^{2} \) |
| 7 | \( 1 - 923.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.00e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.43e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.81e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.32e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.58e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.09e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.48e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.50e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.47e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.29e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.76e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.39e6T + 8.07e13T^{2} \) |
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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
−13.36300209978257812339758450834, −12.21416189583209140163240942646, −11.39798916847575076296404410874, −10.06500833751344015182128565801, −8.020511637206834625388414480945, −7.04678436717967394993162538530, −5.35790075949097753706953542766, −4.57567679492180502566003120050, −3.06864374172009820563395935715, −1.92877385055314760706352215132,
1.92877385055314760706352215132, 3.06864374172009820563395935715, 4.57567679492180502566003120050, 5.35790075949097753706953542766, 7.04678436717967394993162538530, 8.020511637206834625388414480945, 10.06500833751344015182128565801, 11.39798916847575076296404410874, 12.21416189583209140163240942646, 13.36300209978257812339758450834
