| L(s) = 1 | + 3.56·2-s − 27·3-s − 115.·4-s − 210.·5-s − 96.3·6-s + 1.54e3·7-s − 868.·8-s + 729·9-s − 750.·10-s − 1.82e3·11-s + 3.11e3·12-s − 1.23e4·13-s + 5.51e3·14-s + 5.67e3·15-s + 1.16e4·16-s − 3.60e4·17-s + 2.60e3·18-s + 1.82e4·19-s + 2.42e4·20-s − 4.16e4·21-s − 6.49e3·22-s − 9.97e4·23-s + 2.34e4·24-s − 3.39e4·25-s − 4.41e4·26-s − 1.96e4·27-s − 1.78e5·28-s + ⋯ |
| L(s) = 1 | + 0.315·2-s − 0.577·3-s − 0.900·4-s − 0.752·5-s − 0.182·6-s + 1.70·7-s − 0.599·8-s + 0.333·9-s − 0.237·10-s − 0.412·11-s + 0.519·12-s − 1.56·13-s + 0.536·14-s + 0.434·15-s + 0.711·16-s − 1.77·17-s + 0.105·18-s + 0.610·19-s + 0.677·20-s − 0.982·21-s − 0.130·22-s − 1.71·23-s + 0.346·24-s − 0.434·25-s − 0.492·26-s − 0.192·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8775409307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8775409307\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 3.56T + 128T^{2} \) |
| 5 | \( 1 + 210.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.54e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.23e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.60e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.52e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.78e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.15e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.66e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.26e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.67e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.52e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.54e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.17e7T + 8.07e13T^{2} \) |
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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
−11.65642148763858876955277657125, −10.57392392364465662835930399593, −9.372471339496776648195553919854, −8.120197036000921813583751925477, −7.52011500852097348649260517601, −5.76731091937095381704244399834, −4.61138129732780102837082068112, −4.33594940297041218513585019183, −2.23569727826342286127491536366, −0.49463714486490465142525450717,
0.49463714486490465142525450717, 2.23569727826342286127491536366, 4.33594940297041218513585019183, 4.61138129732780102837082068112, 5.76731091937095381704244399834, 7.52011500852097348649260517601, 8.120197036000921813583751925477, 9.372471339496776648195553919854, 10.57392392364465662835930399593, 11.65642148763858876955277657125
