| L(s) = 1 | − 2·4-s − 16·13-s − 3·16-s − 12·19-s − 14·25-s − 20·31-s − 4·37-s + 32·52-s − 8·61-s + 12·64-s − 40·73-s + 24·76-s − 16·79-s − 16·97-s + 28·100-s − 20·103-s − 12·109-s − 6·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s − 4.43·13-s − 3/4·16-s − 2.75·19-s − 2.79·25-s − 3.59·31-s − 0.657·37-s + 4.43·52-s − 1.02·61-s + 3/2·64-s − 4.68·73-s + 2.75·76-s − 1.80·79-s − 1.62·97-s + 14/5·100-s − 1.97·103-s − 1.14·109-s − 0.545·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 14 T^{2} + 97 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 6 T^{2} + 233 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 48 T^{2} + 1082 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 45 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 2 T^{2} + 1057 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 1270 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 138 T^{2} + 7961 T^{4} + 138 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 56 T^{2} + 5130 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 5942 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 224 T^{2} + 19498 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 10 T^{2} + 5305 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 20 T + 244 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 116 T^{2} + 5590 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 42 T^{2} - 8359 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.37985448250584522310984807935, −7.26366739397076294346842574272, −6.93977820165238307002727822344, −6.69711853254314787831053345103, −6.51982598485389474544249610678, −6.23034990845379986738051711789, −5.88744139215323439126040784013, −5.62863390582758952854802152698, −5.58163377645430555648306357782, −5.28006968177684410322761131155, −5.13199469871574049431615957105, −4.70903965527303453290667872732, −4.69224501510124695980253204726, −4.39300273623000433562928488120, −4.08378146346620610223171600047, −4.03308990378162230604245431236, −3.98003074569027842592982941029, −3.41970452538779992965371569839, −3.02191970228723543693745621284, −2.71889605492296418850657063151, −2.48449787143840382279671479400, −2.29864321970376180364209955465, −1.95951074382728026937385784935, −1.70267496650512271625569729760, −1.55390764973579340074370716357, 0, 0, 0, 0,
1.55390764973579340074370716357, 1.70267496650512271625569729760, 1.95951074382728026937385784935, 2.29864321970376180364209955465, 2.48449787143840382279671479400, 2.71889605492296418850657063151, 3.02191970228723543693745621284, 3.41970452538779992965371569839, 3.98003074569027842592982941029, 4.03308990378162230604245431236, 4.08378146346620610223171600047, 4.39300273623000433562928488120, 4.69224501510124695980253204726, 4.70903965527303453290667872732, 5.13199469871574049431615957105, 5.28006968177684410322761131155, 5.58163377645430555648306357782, 5.62863390582758952854802152698, 5.88744139215323439126040784013, 6.23034990845379986738051711789, 6.51982598485389474544249610678, 6.69711853254314787831053345103, 6.93977820165238307002727822344, 7.26366739397076294346842574272, 7.37985448250584522310984807935
