| L(s) = 1 | + 8·2-s + 13.5·3-s + 64·4-s − 206.·5-s + 108.·6-s − 1.55e3·7-s + 512·8-s − 2.00e3·9-s − 1.65e3·10-s − 826.·11-s + 866.·12-s − 1.27e4·13-s − 1.24e4·14-s − 2.80e3·15-s + 4.09e3·16-s + 3.25e4·17-s − 1.60e4·18-s − 5.80e4·19-s − 1.32e4·20-s − 2.10e4·21-s − 6.61e3·22-s − 108.·23-s + 6.93e3·24-s − 3.53e4·25-s − 1.01e5·26-s − 5.67e4·27-s − 9.92e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.289·3-s + 0.5·4-s − 0.740·5-s + 0.204·6-s − 1.70·7-s + 0.353·8-s − 0.916·9-s − 0.523·10-s − 0.187·11-s + 0.144·12-s − 1.60·13-s − 1.20·14-s − 0.214·15-s + 0.250·16-s + 1.60·17-s − 0.647·18-s − 1.94·19-s − 0.370·20-s − 0.494·21-s − 0.132·22-s − 0.00186·23-s + 0.102·24-s − 0.452·25-s − 1.13·26-s − 0.554·27-s − 0.854·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7033716333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7033716333\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 - 13.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 206.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.55e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 826.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.25e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 108.T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.48e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.58e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.92e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.78e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.68e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.35e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.49e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.40e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.21e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.97e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.02e6T + 8.07e13T^{2} \) |
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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
−9.937248080429228287156939567499, −8.704636934418600274310861084284, −7.80283273108565267581732450180, −6.86482220922954964779493722181, −6.05201570148342279713416389915, −5.02459763050559563540063172551, −3.80501122964181138781046899621, −3.11382718845917595773140992636, −2.31062949533681018119714304327, −0.29301971281268955295600507413,
0.29301971281268955295600507413, 2.31062949533681018119714304327, 3.11382718845917595773140992636, 3.80501122964181138781046899621, 5.02459763050559563540063172551, 6.05201570148342279713416389915, 6.86482220922954964779493722181, 7.80283273108565267581732450180, 8.704636934418600274310861084284, 9.937248080429228287156939567499
