| L(s) = 1 | − 423.·5-s + 1.22e3·7-s − 2.78e3·11-s + 3.72e3·13-s + 1.16e4·17-s − 4.51e4·19-s − 2.77e4·23-s + 1.01e5·25-s + 2.10e5·29-s + 1.72e5·31-s − 5.20e5·35-s − 2.69e5·37-s + 3.79e5·41-s + 1.03e6·43-s + 6.24e5·47-s + 6.86e5·49-s − 1.84e6·53-s + 1.17e6·55-s − 2.74e6·59-s − 2.70e6·61-s − 1.57e6·65-s + 9.89e5·67-s − 3.30e6·71-s − 1.10e6·73-s − 3.41e6·77-s − 8.02e6·79-s + 3.55e6·83-s + ⋯ |
| L(s) = 1 | − 1.51·5-s + 1.35·7-s − 0.630·11-s + 0.470·13-s + 0.574·17-s − 1.51·19-s − 0.475·23-s + 1.29·25-s + 1.60·29-s + 1.04·31-s − 2.05·35-s − 0.874·37-s + 0.860·41-s + 1.98·43-s + 0.876·47-s + 0.833·49-s − 1.69·53-s + 0.954·55-s − 1.74·59-s − 1.52·61-s − 0.713·65-s + 0.401·67-s − 1.09·71-s − 0.331·73-s − 0.853·77-s − 1.83·79-s + 0.682·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 423.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.22e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.78e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.72e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.51e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.69e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.03e6T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.84e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.70e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.89e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.02e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.07e6T + 8.07e13T^{2} \) |
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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
−10.53707554512486735537847161295, −8.795775551520547517033475185632, −8.051380703278408217552028101814, −7.59459950283798577630923035152, −6.16235475002336461958525948902, −4.71828095180028270949401433133, −4.16409846189243188436603890318, −2.75757607589599738985313196226, −1.25475356905918258292320794011, 0,
1.25475356905918258292320794011, 2.75757607589599738985313196226, 4.16409846189243188436603890318, 4.71828095180028270949401433133, 6.16235475002336461958525948902, 7.59459950283798577630923035152, 8.051380703278408217552028101814, 8.795775551520547517033475185632, 10.53707554512486735537847161295
