L(s) = 1 | + 5·2-s + 8·4-s − 5·8-s + 27·9-s + 68·11-s − 25·16-s + 135·18-s + 340·22-s + 40·23-s + 125·25-s − 332·29-s − 40·32-s + 216·36-s − 450·37-s − 360·43-s + 544·44-s + 200·46-s + 625·50-s − 590·53-s − 1.66e3·58-s − 487·64-s + 740·67-s + 1.37e3·71-s − 135·72-s − 2.25e3·74-s + 1.38e3·79-s − 1.80e3·86-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 4-s − 0.220·8-s + 9-s + 1.86·11-s − 0.390·16-s + 1.76·18-s + 3.29·22-s + 0.362·23-s + 25-s − 2.12·29-s − 0.220·32-s + 36-s − 1.99·37-s − 1.27·43-s + 1.86·44-s + 0.641·46-s + 1.76·50-s − 1.52·53-s − 3.75·58-s − 0.951·64-s + 1.34·67-s + 2.30·71-s − 0.220·72-s − 3.53·74-s + 1.97·79-s − 2.25·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.968010391\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.968010391\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 68 T + 3293 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T - 10567 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 450 T + 151847 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 590 T + 199223 T^{2} + 590 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 740 T + 246837 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 688 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1384 T + 1422417 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−15.58150961034859080593446467880, −14.56790425228937279716035732933, −14.26829920098738522370876216374, −13.71392684831262013852724372012, −13.14577009457773151276223702368, −12.57030356794099798553145416739, −12.35158545687795217665964664544, −11.52360495984211484578701326346, −10.99618450099550377412016358504, −10.09827857541056548882226540943, −9.306031528502591944840417068787, −8.935841680855088529441486642920, −7.79471782090051441495854058209, −6.80492720372833074655131100255, −6.51814749140058946600446650259, −5.32295841663858738679933007277, −4.84578728785744800676254557824, −3.79768215730231864999750578192, −3.61191549279543295011614079246, −1.61408681898428496179850998824,
1.61408681898428496179850998824, 3.61191549279543295011614079246, 3.79768215730231864999750578192, 4.84578728785744800676254557824, 5.32295841663858738679933007277, 6.51814749140058946600446650259, 6.80492720372833074655131100255, 7.79471782090051441495854058209, 8.935841680855088529441486642920, 9.306031528502591944840417068787, 10.09827857541056548882226540943, 10.99618450099550377412016358504, 11.52360495984211484578701326346, 12.35158545687795217665964664544, 12.57030356794099798553145416739, 13.14577009457773151276223702368, 13.71392684831262013852724372012, 14.26829920098738522370876216374, 14.56790425228937279716035732933, 15.58150961034859080593446467880