| L(s) = 1 | + 3.09·2-s − 27·3-s − 118.·4-s − 156.·5-s − 83.6·6-s + 11.3·7-s − 763.·8-s + 729·9-s − 484.·10-s + 4.29e3·11-s + 3.19e3·12-s + 1.74e3·13-s + 35.1·14-s + 4.22e3·15-s + 1.27e4·16-s − 4.76e3·17-s + 2.25e3·18-s + 1.82e4·19-s + 1.85e4·20-s − 306.·21-s + 1.33e4·22-s + 8.20e4·23-s + 2.06e4·24-s − 5.36e4·25-s + 5.39e3·26-s − 1.96e4·27-s − 1.34e3·28-s + ⋯ |
| L(s) = 1 | + 0.273·2-s − 0.577·3-s − 0.925·4-s − 0.559·5-s − 0.158·6-s + 0.0125·7-s − 0.527·8-s + 0.333·9-s − 0.153·10-s + 0.973·11-s + 0.534·12-s + 0.219·13-s + 0.00342·14-s + 0.323·15-s + 0.780·16-s − 0.235·17-s + 0.0912·18-s + 0.611·19-s + 0.517·20-s − 0.00722·21-s + 0.266·22-s + 1.40·23-s + 0.304·24-s − 0.686·25-s + 0.0601·26-s − 0.192·27-s − 0.0115·28-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 3.09T + 128T^{2} \) |
| 5 | \( 1 + 156.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 11.3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.29e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.74e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.76e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.82e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.77e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.50e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.69e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.91e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.30e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.56e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 9.37e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.38e7T + 8.07e13T^{2} \) |
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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
−11.16953475043893414853271257182, −9.771686349446360176908129671340, −8.983532623056283486374567774693, −7.79862131369422093725100431143, −6.55134447206579323376188950155, −5.37411992641858542628532293745, −4.34826128790445902038562212085, −3.39745875161129907913183025293, −1.21660521409085229530038292975, 0,
1.21660521409085229530038292975, 3.39745875161129907913183025293, 4.34826128790445902038562212085, 5.37411992641858542628532293745, 6.55134447206579323376188950155, 7.79862131369422093725100431143, 8.983532623056283486374567774693, 9.771686349446360176908129671340, 11.16953475043893414853271257182
