| L(s) = 1 | − 8·2-s − 89.7·3-s + 64·4-s − 125·5-s + 718.·6-s + 78.5·7-s − 512·8-s + 5.87e3·9-s + 1.00e3·10-s − 509.·11-s − 5.74e3·12-s − 8.64e3·13-s − 628.·14-s + 1.12e4·15-s + 4.09e3·16-s − 3.24e4·17-s − 4.69e4·18-s − 2.09e4·19-s − 8.00e3·20-s − 7.04e3·21-s + 4.07e3·22-s + 2.52e4·23-s + 4.59e4·24-s + 1.56e4·25-s + 6.91e4·26-s − 3.31e5·27-s + 5.02e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.91·3-s + 0.5·4-s − 0.447·5-s + 1.35·6-s + 0.0865·7-s − 0.353·8-s + 2.68·9-s + 0.316·10-s − 0.115·11-s − 0.959·12-s − 1.09·13-s − 0.0611·14-s + 0.858·15-s + 0.250·16-s − 1.59·17-s − 1.89·18-s − 0.699·19-s − 0.223·20-s − 0.166·21-s + 0.0815·22-s + 0.432·23-s + 0.678·24-s + 0.199·25-s + 0.771·26-s − 3.23·27-s + 0.0432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 8T \) |
| 5 | \( 1 + 125T \) |
| 31 | \( 1 - 2.97e4T \) |
| good | 3 | \( 1 + 89.7T + 2.18e3T^{2} \) |
| 7 | \( 1 - 78.5T + 8.23e5T^{2} \) |
| 11 | \( 1 + 509.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.64e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.09e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.69e4T + 1.72e10T^{2} \) |
| 37 | \( 1 - 3.70e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.11e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.25e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.63e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.74e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.92e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.44e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.19e5T + 8.07e13T^{2} \) |
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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
−10.24402056713349332176983794792, −9.300123120005368297074454736180, −7.907550831078642799956146648500, −6.90289331496643545131931934246, −6.31738976340749762866149444332, −5.04706430408812849206071296041, −4.29746999832155717623803290732, −2.23362009122687125139139122543, −0.798499402436378994240514405169, 0,
0.798499402436378994240514405169, 2.23362009122687125139139122543, 4.29746999832155717623803290732, 5.04706430408812849206071296041, 6.31738976340749762866149444332, 6.90289331496643545131931934246, 7.907550831078642799956146648500, 9.300123120005368297074454736180, 10.24402056713349332176983794792
