| L(s) = 1 | − 12·5-s + 12·17-s + 12·19-s + 60·23-s + 66·25-s + 68·31-s − 240·47-s + 8·49-s − 204·53-s − 196·61-s − 180·79-s + 108·83-s − 144·85-s − 144·95-s − 144·107-s − 76·109-s − 48·113-s − 720·115-s + 96·121-s − 156·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 816·155-s + ⋯ |
| L(s) = 1 | − 2.39·5-s + 0.705·17-s + 0.631·19-s + 2.60·23-s + 2.63·25-s + 2.19·31-s − 5.10·47-s + 8/49·49-s − 3.84·53-s − 3.21·61-s − 2.27·79-s + 1.30·83-s − 1.69·85-s − 1.51·95-s − 1.34·107-s − 0.697·109-s − 0.424·113-s − 6.26·115-s + 0.793·121-s − 1.24·125-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 − 5.26·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4806187608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4806187608\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 - 8 T^{2} - 3894 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 96 T^{2} + 29786 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 352 T^{2} + 71898 T^{4} - 352 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 489 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 713 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 30 T + 891 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1096 T^{2} + 1242474 T^{4} - 1096 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 34 T + 1761 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 5352 T^{2} + 12407498 T^{4} - 5352 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 5672 T^{2} + 14203050 T^{4} - 5672 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 120 T + 7626 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 102 T + 8057 T^{2} + 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 10848 T^{2} + 53616410 T^{4} - 10848 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 98 T + 8691 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 6508 T^{2} + 32310150 T^{4} + 6508 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 90 T + 6569 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 54 T + 7779 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 19172 T^{2} + 267913158 T^{4} - 19172 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.26186041706497531726474535181, −6.12310179363891316749945228676, −5.88193141609136567394916415885, −5.69992144826141595842950022742, −5.14285688501537922992111360917, −5.09574616700765144850903279678, −5.00300765022317042222633314926, −4.55166912381640957759594387508, −4.52661808513471474313379342374, −4.45082910982069498577606907041, −4.42926075245793265168381229551, −3.62531406683303089254040972586, −3.45672483929030347647434243328, −3.43782493273158093036393690539, −3.30605098114644118486140066233, −3.02479279264904845119846223968, −2.87421408837980613356708234999, −2.58300709810755856131200400913, −2.17812806087464083816625959186, −1.63963185506767959032358117165, −1.26678775294253196977690566246, −1.21338092230257440535752697091, −1.14824841749661257816504019944, −0.25320905450733790111707806014, −0.20545440607039526451634565472,
0.20545440607039526451634565472, 0.25320905450733790111707806014, 1.14824841749661257816504019944, 1.21338092230257440535752697091, 1.26678775294253196977690566246, 1.63963185506767959032358117165, 2.17812806087464083816625959186, 2.58300709810755856131200400913, 2.87421408837980613356708234999, 3.02479279264904845119846223968, 3.30605098114644118486140066233, 3.43782493273158093036393690539, 3.45672483929030347647434243328, 3.62531406683303089254040972586, 4.42926075245793265168381229551, 4.45082910982069498577606907041, 4.52661808513471474313379342374, 4.55166912381640957759594387508, 5.00300765022317042222633314926, 5.09574616700765144850903279678, 5.14285688501537922992111360917, 5.69992144826141595842950022742, 5.88193141609136567394916415885, 6.12310179363891316749945228676, 6.26186041706497531726474535181
