| L(s) = 1 | − 4·2-s + 3-s + 10·4-s − 3·5-s − 4·6-s − 20·8-s + 3·9-s + 12·10-s + 10·12-s − 8·13-s − 3·15-s + 35·16-s − 12·18-s − 30·20-s − 6·23-s − 20·24-s + 5·25-s + 32·26-s + 8·27-s + 12·30-s − 56·32-s + 30·36-s − 8·39-s + 60·40-s + 18·41-s − 9·45-s + 24·46-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 5·4-s − 1.34·5-s − 1.63·6-s − 7.07·8-s + 9-s + 3.79·10-s + 2.88·12-s − 2.21·13-s − 0.774·15-s + 35/4·16-s − 2.82·18-s − 6.70·20-s − 1.25·23-s − 4.08·24-s + 25-s + 6.27·26-s + 1.53·27-s + 2.19·30-s − 9.89·32-s + 5·36-s − 1.28·39-s + 9.48·40-s + 2.81·41-s − 1.34·45-s + 3.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08203690356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08203690356\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 324 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 2184 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 2334 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6366 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 73 T^{2} + 2760 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 205 T^{2} + 18816 T^{4} - 205 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.86312767863944714342216889134, −6.74476466543811615451028628138, −6.60435305875096911166052674978, −6.55435547402494762405103489802, −6.04220996845439553859974218777, −5.69531783763813326335933050794, −5.52923623971073379891182565642, −5.45337106132197789571519096472, −5.02990051690629300453464516254, −4.83383657306777843223296569265, −4.24172254339516635578841387428, −4.19672429070276018582710666226, −4.18297389085895659119969362874, −3.81415785378694642388076338441, −3.54860932470978812693029711503, −3.05536164915501303380330277931, −2.67371401790238602243684260157, −2.63577472445902719088369971702, −2.44751314752646849657416039542, −2.35510228566111763114089670978, −1.68198667446727389748374353355, −1.34030470561268599264958109619, −1.21512007577811582132652547180, −0.64818794872419168179988565399, −0.12481520335432469099098900151,
0.12481520335432469099098900151, 0.64818794872419168179988565399, 1.21512007577811582132652547180, 1.34030470561268599264958109619, 1.68198667446727389748374353355, 2.35510228566111763114089670978, 2.44751314752646849657416039542, 2.63577472445902719088369971702, 2.67371401790238602243684260157, 3.05536164915501303380330277931, 3.54860932470978812693029711503, 3.81415785378694642388076338441, 4.18297389085895659119969362874, 4.19672429070276018582710666226, 4.24172254339516635578841387428, 4.83383657306777843223296569265, 5.02990051690629300453464516254, 5.45337106132197789571519096472, 5.52923623971073379891182565642, 5.69531783763813326335933050794, 6.04220996845439553859974218777, 6.55435547402494762405103489802, 6.60435305875096911166052674978, 6.74476466543811615451028628138, 6.86312767863944714342216889134
