| L(s) = 1 | + 6·9-s + 16·13-s − 4·17-s − 96·29-s + 24·37-s + 132·41-s + 76·49-s + 104·53-s − 216·61-s − 172·73-s + 41·81-s + 220·89-s + 184·97-s − 40·101-s + 24·109-s − 156·113-s + 96·117-s + 278·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 1.23·13-s − 0.235·17-s − 3.31·29-s + 0.648·37-s + 3.21·41-s + 1.55·49-s + 1.96·53-s − 3.54·61-s − 2.35·73-s + 0.506·81-s + 2.47·89-s + 1.89·97-s − 0.396·101-s + 0.220·109-s − 1.38·113-s + 0.820·117-s + 2.29·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.156·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.282623164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282623164\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} - 5 T^{4} - 2 p^{5} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 10 p^{3} T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 278 T^{2} + 47019 T^{4} - 278 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 403 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 246 T^{2} - 80629 T^{4} - 246 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1292 T^{2} + 951654 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 48 T + 1554 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1164 T^{2} + 1957670 T^{4} - 1164 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T - 42 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 66 T + 3747 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5956 T^{2} + 8470 p^{2} T^{4} - 5956 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 4242 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 52 T + 5590 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3556 T^{2} + 13597606 T^{4} - 3556 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 108 T + 9654 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 3930 T^{2} + 21732443 T^{4} + 3930 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 5276 T^{2} + 48951430 T^{4} + 5276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 86 T + 12331 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18764 T^{2} + 161638950 T^{4} - 18764 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 23542 T^{2} + 229874059 T^{4} - 23542 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 110 T + 14467 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.34949569471485177332416249540, −6.22504146325505689365013471229, −6.03763056958176854730109550416, −5.87927635045786217859516496374, −5.83413383390922789639628526071, −5.43091110559648244063156006817, −5.32081690826987536778034976426, −4.87232227557314150049779419073, −4.84386674577686799497718984747, −4.32093595066038586544049326898, −4.23306413947165999513602631285, −4.10700759135977483498209547463, −3.90777609101768801238352124607, −3.62980704464770390131412421267, −3.36253360684305569972331591006, −2.99040471836673280357591594229, −2.95092300557671392960489109723, −2.35293398897215094166517006648, −2.18750107540686540076604736543, −2.07443487440397529041760415575, −1.61681224708405568228813049897, −1.21768446013744379460590147671, −1.10738367023779059958914205301, −0.63954987363462032774216436055, −0.24804790524401347921635521175,
0.24804790524401347921635521175, 0.63954987363462032774216436055, 1.10738367023779059958914205301, 1.21768446013744379460590147671, 1.61681224708405568228813049897, 2.07443487440397529041760415575, 2.18750107540686540076604736543, 2.35293398897215094166517006648, 2.95092300557671392960489109723, 2.99040471836673280357591594229, 3.36253360684305569972331591006, 3.62980704464770390131412421267, 3.90777609101768801238352124607, 4.10700759135977483498209547463, 4.23306413947165999513602631285, 4.32093595066038586544049326898, 4.84386674577686799497718984747, 4.87232227557314150049779419073, 5.32081690826987536778034976426, 5.43091110559648244063156006817, 5.83413383390922789639628526071, 5.87927635045786217859516496374, 6.03763056958176854730109550416, 6.22504146325505689365013471229, 6.34949569471485177332416249540
