| L(s) = 1 | − 6·3-s + 2·4-s + 2·5-s + 10·7-s + 17·9-s − 12·12-s + 8·13-s − 12·15-s − 12·17-s + 4·20-s − 60·21-s − 6·23-s + 25-s − 30·27-s + 20·28-s + 8·31-s + 20·35-s + 34·36-s + 12·37-s − 48·39-s − 28·43-s + 34·45-s − 12·47-s + 61·49-s + 72·51-s + 16·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 4-s + 0.894·5-s + 3.77·7-s + 17/3·9-s − 3.46·12-s + 2.21·13-s − 3.09·15-s − 2.91·17-s + 0.894·20-s − 13.0·21-s − 1.25·23-s + 1/5·25-s − 5.77·27-s + 3.77·28-s + 1.43·31-s + 3.38·35-s + 17/3·36-s + 1.97·37-s − 7.68·39-s − 4.26·43-s + 5.06·45-s − 1.75·47-s + 61/7·49-s + 10.0·51-s + 2.21·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.229631731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229631731\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2463 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 20 T^{2} + 39 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 246 T^{3} + 1428 T^{4} + 246 p T^{5} + 53 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 38 T^{2} + 1443 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 8 T^{2} - 80 T^{3} + 2239 T^{4} - 80 p T^{5} - 8 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 4707 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 3 p T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 164 T^{2} - 1392 T^{3} + 13191 T^{4} - 1392 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 128 T^{2} + 960 T^{3} + 5751 T^{4} + 960 p T^{5} + 128 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 79 T^{2} + 70 T^{3} - 1988 T^{4} + 70 p T^{5} + 79 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 368 T^{2} + 4224 T^{3} + 39663 T^{4} + 4224 p T^{5} + 368 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T + 200 T^{2} - 1824 T^{3} + 20655 T^{4} - 1824 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 178 T^{2} + 16299 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 281 T^{2} - 3114 T^{3} + 31620 T^{4} - 3114 p T^{5} + 281 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 27462 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.619261602319025562529121165025, −8.493897877799311899413935143154, −8.070569266344549656308283504342, −7.82681331599712041657470485813, −7.62497987485950307595613875208, −7.05438092233608930529434958414, −6.99699499171948781973618803398, −6.59914667291269252852283189549, −6.26970709892000490355201848940, −6.22363257781224593399191038692, −5.97536995224891624199793017354, −5.87777112152146549796695598733, −5.49838008688405506348044011872, −5.03674933363583516986998518936, −4.91939563703591071652935929127, −4.61924441858115855013987352411, −4.60598100220317454596140416955, −4.24514754222682582703660249740, −3.79905384661436785738853675775, −3.12186617650362319691617646812, −2.25465609838410633543590467626, −2.08550631939329448872035821156, −1.63655223430770615177351639167, −1.51120056911548877095012953310, −0.70659462693298453991776413315,
0.70659462693298453991776413315, 1.51120056911548877095012953310, 1.63655223430770615177351639167, 2.08550631939329448872035821156, 2.25465609838410633543590467626, 3.12186617650362319691617646812, 3.79905384661436785738853675775, 4.24514754222682582703660249740, 4.60598100220317454596140416955, 4.61924441858115855013987352411, 4.91939563703591071652935929127, 5.03674933363583516986998518936, 5.49838008688405506348044011872, 5.87777112152146549796695598733, 5.97536995224891624199793017354, 6.22363257781224593399191038692, 6.26970709892000490355201848940, 6.59914667291269252852283189549, 6.99699499171948781973618803398, 7.05438092233608930529434958414, 7.62497987485950307595613875208, 7.82681331599712041657470485813, 8.070569266344549656308283504342, 8.493897877799311899413935143154, 8.619261602319025562529121165025
