| L(s) = 1 | + 15·4-s − 18·9-s + 44·11-s + 49·16-s + 108·19-s − 444·29-s − 80·31-s − 270·36-s − 988·41-s + 660·44-s + 1.30e3·49-s − 784·59-s − 2.20e3·61-s − 945·64-s + 912·71-s + 1.62e3·76-s + 460·79-s + 243·81-s − 1.94e3·89-s − 792·99-s − 548·101-s − 2.20e3·109-s − 6.66e3·116-s + 1.21e3·121-s − 1.20e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 2/3·9-s + 1.20·11-s + 0.765·16-s + 1.30·19-s − 2.84·29-s − 0.463·31-s − 5/4·36-s − 3.76·41-s + 2.26·44-s + 3.80·49-s − 1.72·59-s − 4.63·61-s − 1.84·64-s + 1.52·71-s + 2.44·76-s + 0.655·79-s + 1/3·81-s − 2.31·89-s − 0.804·99-s − 0.539·101-s − 1.93·109-s − 5.33·116-s + 0.909·121-s − 0.869·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.950743670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.950743670\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 15 T^{2} + 11 p^{4} T^{4} - 15 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1304 T^{2} + 660270 T^{4} - 1304 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4844 T^{2} + 12469974 T^{4} - 4844 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4464 T^{2} + 48237662 T^{4} - 4464 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 54 T + 11774 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 11790 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 222 T + 43642 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 169516 T^{2} + 12278571894 T^{4} - 169516 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 193816 T^{2} + 21768096510 T^{4} - 193816 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 375228 T^{2} + 56679586886 T^{4} - 375228 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 321644 T^{2} + 69238692534 T^{4} - 321644 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 1104 T + 736358 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 391244 T^{2} + 55036171350 T^{4} - 391244 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 456 T + 488494 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 333020 T^{2} + 146783139366 T^{4} - 333020 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 230 T + 954126 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 494492 T^{2} + 610128969846 T^{4} - 494492 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 972 T + 1645606 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 39164 T^{2} - 373793572218 T^{4} - 39164 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.88689209894152707896240179699, −6.86711223117972757708025309324, −6.70884545013897300925619159669, −6.13239770811674301740016979319, −5.97447999526733709883579303045, −5.88261950766662318032930445976, −5.70820420761944387017170179144, −5.32405005264839667475787472954, −5.18784400265825709432963381973, −4.76844105866890612281669219017, −4.67801349241768296738920585915, −4.05685286102809497008716435209, −4.04499045967386209324665594763, −3.62304615368402697042245789177, −3.53697824011665292713893105777, −3.06197791381191993155869170195, −2.96504690377745871060080933498, −2.59743699604936556913094328077, −2.41796061896203910475026284751, −1.76535335782340461396077217258, −1.70100711505853940977267372537, −1.58808755750868779990321346023, −1.29115644146394651132330971686, −0.43933794673993289051398098611, −0.32642839023806344541901065787,
0.32642839023806344541901065787, 0.43933794673993289051398098611, 1.29115644146394651132330971686, 1.58808755750868779990321346023, 1.70100711505853940977267372537, 1.76535335782340461396077217258, 2.41796061896203910475026284751, 2.59743699604936556913094328077, 2.96504690377745871060080933498, 3.06197791381191993155869170195, 3.53697824011665292713893105777, 3.62304615368402697042245789177, 4.04499045967386209324665594763, 4.05685286102809497008716435209, 4.67801349241768296738920585915, 4.76844105866890612281669219017, 5.18784400265825709432963381973, 5.32405005264839667475787472954, 5.70820420761944387017170179144, 5.88261950766662318032930445976, 5.97447999526733709883579303045, 6.13239770811674301740016979319, 6.70884545013897300925619159669, 6.86711223117972757708025309324, 6.88689209894152707896240179699
