| L(s) = 1 | − 2.18e3·3-s + 7.76e4·5-s − 7.62e5·7-s + 4.78e6·9-s − 4.80e7·11-s + 2.85e8·13-s − 1.69e8·15-s − 3.17e9·17-s + 5.89e9·19-s + 1.66e9·21-s + 3.33e8·23-s − 2.44e10·25-s − 1.04e10·27-s + 1.17e11·29-s + 2.25e11·31-s + 1.05e11·33-s − 5.91e10·35-s − 4.77e11·37-s − 6.23e11·39-s + 1.19e12·41-s − 1.06e12·43-s + 3.71e11·45-s − 1.32e12·47-s − 4.16e12·49-s + 6.94e12·51-s − 6.57e12·53-s − 3.72e12·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.444·5-s − 0.349·7-s + 1/3·9-s − 0.742·11-s + 1.26·13-s − 0.256·15-s − 1.87·17-s + 1.51·19-s + 0.201·21-s + 0.0203·23-s − 0.802·25-s − 0.192·27-s + 1.26·29-s + 1.47·31-s + 0.428·33-s − 0.155·35-s − 0.827·37-s − 0.727·39-s + 0.959·41-s − 0.598·43-s + 0.148·45-s − 0.381·47-s − 0.877·49-s + 1.08·51-s − 0.768·53-s − 0.330·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{7} T \) |
| good | 5 | \( 1 - 77646 T + p^{15} T^{2} \) |
| 7 | \( 1 + 108872 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 4364652 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 21933086 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 3173671566 T + p^{15} T^{2} \) |
| 19 | \( 1 - 5895116260 T + p^{15} T^{2} \) |
| 23 | \( 1 - 333010392 T + p^{15} T^{2} \) |
| 29 | \( 1 - 117285392310 T + p^{15} T^{2} \) |
| 31 | \( 1 - 225821452768 T + p^{15} T^{2} \) |
| 37 | \( 1 + 477657973906 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1196721561882 T + p^{15} T^{2} \) |
| 43 | \( 1 + 1066802913668 T + p^{15} T^{2} \) |
| 47 | \( 1 + 1324913565264 T + p^{15} T^{2} \) |
| 53 | \( 1 + 6573181204962 T + p^{15} T^{2} \) |
| 59 | \( 1 + 7973946241140 T + p^{15} T^{2} \) |
| 61 | \( 1 - 14311350203222 T + p^{15} T^{2} \) |
| 67 | \( 1 + 41052380998124 T + p^{15} T^{2} \) |
| 71 | \( 1 + 67253761134072 T + p^{15} T^{2} \) |
| 73 | \( 1 + 156200366359942 T + p^{15} T^{2} \) |
| 79 | \( 1 - 138004701018640 T + p^{15} T^{2} \) |
| 83 | \( 1 + 469396029824988 T + p^{15} T^{2} \) |
| 89 | \( 1 + 422649074576790 T + p^{15} T^{2} \) |
| 97 | \( 1 + 201862519502686 T + p^{15} T^{2} \) |
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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
−11.77779513075998299100610505690, −10.76254722286428153721772194916, −9.673875121034417987814118876192, −8.341736625551158393572238865359, −6.76984612063547167331601635495, −5.81168845184723670908544602621, −4.50897025015523295578355838676, −2.89934277523136363857829137799, −1.37217043494279699346093884733, 0,
1.37217043494279699346093884733, 2.89934277523136363857829137799, 4.50897025015523295578355838676, 5.81168845184723670908544602621, 6.76984612063547167331601635495, 8.341736625551158393572238865359, 9.673875121034417987814118876192, 10.76254722286428153721772194916, 11.77779513075998299100610505690
