| L(s) = 1 | − 6·9-s − 16·13-s − 48·17-s + 10·25-s − 48·29-s + 160·37-s + 168·41-s + 52·49-s + 192·53-s − 344·61-s + 88·73-s + 27·81-s − 264·89-s − 344·97-s − 720·101-s + 296·109-s − 96·113-s + 96·117-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 288·153-s + 157-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 1.23·13-s − 2.82·17-s + 2/5·25-s − 1.65·29-s + 4.32·37-s + 4.09·41-s + 1.06·49-s + 3.62·53-s − 5.63·61-s + 1.20·73-s + 1/3·81-s − 2.96·89-s − 3.54·97-s − 7.12·101-s + 2.71·109-s − 0.849·113-s + 0.820·117-s + 2.80·121-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 + 1.88·153-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5700164813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5700164813\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2598 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 174 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 24 T + 542 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 580 T^{2} + 160422 T^{4} - 580 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1540 T^{2} + 1106502 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 206 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2980 T^{2} + 3882822 T^{4} - 2980 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 80 T + 4158 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 + 380 T^{2} + 6136422 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6532 T^{2} + 19688838 T^{4} - 6532 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 96 T + 6302 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10900 T^{2} + 53588742 T^{4} - 10900 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 172 T + 14118 T^{2} + 172 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12100 T^{2} + 70269222 T^{4} - 12100 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 3452 T^{2} + 51544518 T^{4} + 3452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 44 T + 4662 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12242 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9412 T^{2} + 57343398 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 132 T + 19478 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 172 T + 23334 T^{2} + 172 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.08380043608766701081396600544, −6.82990619238728316829601696903, −6.47229191228138885803701841909, −6.35677202762309094159338886679, −5.87943616792729111257135389943, −5.81123025807721021414199676933, −5.78268757241104367398593736474, −5.38501580374322939508690597641, −5.36174605788941575947212361576, −4.55796694690044616547228428850, −4.55791846992699917878173957586, −4.35920078677987483733576509083, −4.31412000476375379144784929119, −3.95652878401445413172882267575, −3.92022531696213587174409878265, −3.10450763353090454235120502117, −2.88904211906844145716978552487, −2.66353838255998488157299159541, −2.51774972179957770141224250289, −2.29264751971365362972600347114, −2.08003386669323788217641814262, −1.30001547705189447048895020778, −1.19013605203278992811796027310, −0.58515600326473578932816082798, −0.14573445633586370563276918944,
0.14573445633586370563276918944, 0.58515600326473578932816082798, 1.19013605203278992811796027310, 1.30001547705189447048895020778, 2.08003386669323788217641814262, 2.29264751971365362972600347114, 2.51774972179957770141224250289, 2.66353838255998488157299159541, 2.88904211906844145716978552487, 3.10450763353090454235120502117, 3.92022531696213587174409878265, 3.95652878401445413172882267575, 4.31412000476375379144784929119, 4.35920078677987483733576509083, 4.55791846992699917878173957586, 4.55796694690044616547228428850, 5.36174605788941575947212361576, 5.38501580374322939508690597641, 5.78268757241104367398593736474, 5.81123025807721021414199676933, 5.87943616792729111257135389943, 6.35677202762309094159338886679, 6.47229191228138885803701841909, 6.82990619238728316829601696903, 7.08380043608766701081396600544
