| L(s) = 1 | + 14·3-s − 70·5-s − 23·9-s − 62·11-s − 1.82e3·13-s − 980·15-s − 1.69e3·17-s + 826·19-s + 2.73e3·23-s + 2.48e3·25-s + 14·27-s − 2.85e3·29-s − 2.67e3·31-s − 868·33-s − 9.14e3·37-s − 2.54e4·39-s − 6.13e3·41-s + 1.60e4·43-s + 1.61e3·45-s − 2.53e4·47-s − 2.37e4·51-s + 1.49e4·53-s + 4.34e3·55-s + 1.15e4·57-s − 1.10e3·59-s + 2.80e4·61-s + 1.27e5·65-s + ⋯ |
| L(s) = 1 | + 0.898·3-s − 1.25·5-s − 0.0946·9-s − 0.154·11-s − 2.98·13-s − 1.12·15-s − 1.42·17-s + 0.524·19-s + 1.07·23-s + 0.793·25-s + 0.00369·27-s − 0.629·29-s − 0.499·31-s − 0.138·33-s − 1.09·37-s − 2.68·39-s − 0.569·41-s + 1.32·43-s + 0.118·45-s − 1.67·47-s − 1.27·51-s + 0.731·53-s + 0.193·55-s + 0.471·57-s − 0.0413·59-s + 0.964·61-s + 3.74·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 73 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 p T + 2419 T^{2} + 14 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 62 T + 307579 T^{2} + 62 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 140 p T + 1508750 T^{2} + 140 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1694 T + 3404179 T^{2} + 1694 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 826 T + 4642131 T^{2} - 826 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2734 T + 10266499 T^{2} - 2734 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2852 T + 25156270 T^{2} + 2852 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2674 T + 58089971 T^{2} + 2674 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9146 T + 154583427 T^{2} + 9146 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6132 T + 240555334 T^{2} + 6132 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16040 T + 215636742 T^{2} - 16040 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 25326 T + 616649779 T^{2} + 25326 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14958 T + 543936427 T^{2} - 14958 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1106 T + 1400371723 T^{2} + 1106 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 28042 T + 1885112387 T^{2} - 28042 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 102642 T + 5264587579 T^{2} - 102642 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11056 T + 3430664782 T^{2} - 11056 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 35070 T + 3668746195 T^{2} + 35070 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 101762 T + 5366098883 T^{2} + 101762 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 44632 T + 2269443286 T^{2} + 44632 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 75474 T + 9250340803 T^{2} - 75474 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8316 T + 16824750934 T^{2} - 8316 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.273561886281924382060755942869, −8.938540847467628183797002325972, −8.335010838052384151052155629658, −8.175904435272867054406961562599, −7.46402836922815541506951580555, −7.36140185955878622454738884860, −6.84471393946814755125016165193, −6.68243769477460703064187798288, −5.39334509427887104673177419579, −5.38868077315536056409838861131, −4.60682136051758522715473113066, −4.50461262193567819874454590575, −3.69364195861943475265032937162, −3.29265580862890299875314060897, −2.66194614803501803473022161238, −2.40304204886802647435093449984, −1.83673659774530021222871226065, −0.792786070215387526813688467770, 0, 0,
0.792786070215387526813688467770, 1.83673659774530021222871226065, 2.40304204886802647435093449984, 2.66194614803501803473022161238, 3.29265580862890299875314060897, 3.69364195861943475265032937162, 4.50461262193567819874454590575, 4.60682136051758522715473113066, 5.38868077315536056409838861131, 5.39334509427887104673177419579, 6.68243769477460703064187798288, 6.84471393946814755125016165193, 7.36140185955878622454738884860, 7.46402836922815541506951580555, 8.175904435272867054406961562599, 8.335010838052384151052155629658, 8.938540847467628183797002325972, 9.273561886281924382060755942869
