| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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
−12.08364468993422966725803913365, −11.02535680697844516023198077478, −10.15882383606134226547673514760, −9.271961104241637535545239229785, −8.040168997743132885524454401343, −6.88739988712124694910513494567, −5.83141960118659190009933936685, −3.08140588177030148191920135805, −1.23475592003302044560197847390, 0,
1.23475592003302044560197847390, 3.08140588177030148191920135805, 5.83141960118659190009933936685, 6.88739988712124694910513494567, 8.040168997743132885524454401343, 9.271961104241637535545239229785, 10.15882383606134226547673514760, 11.02535680697844516023198077478, 12.08364468993422966725803913365
