| L(s) = 1 | − 2-s − 2·3-s + 3·4-s + 2·6-s + 10·7-s − 4·8-s + 7·9-s − 6·12-s − 10·13-s − 10·14-s + 6·16-s − 2·17-s − 7·18-s − 9·19-s − 20·21-s + 8·23-s + 8·24-s + 5·25-s + 10·26-s − 22·27-s + 30·28-s − 18·29-s − 11·31-s − 6·32-s + 2·34-s + 21·36-s + 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 3/2·4-s + 0.816·6-s + 3.77·7-s − 1.41·8-s + 7/3·9-s − 1.73·12-s − 2.77·13-s − 2.67·14-s + 3/2·16-s − 0.485·17-s − 1.64·18-s − 2.06·19-s − 4.36·21-s + 1.66·23-s + 1.63·24-s + 25-s + 1.96·26-s − 4.23·27-s + 5.66·28-s − 3.34·29-s − 1.97·31-s − 1.06·32-s + 0.342·34-s + 7/2·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7131096239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7131096239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T - p T^{2} - T^{3} + 3 T^{4} - p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 26 T^{2} - 8 T^{3} + 543 T^{4} - 8 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T + 34 T^{2} + 81 T^{3} + 309 T^{4} + 81 p T^{5} + 34 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 88 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 11 T + 30 T^{2} + 319 T^{3} + 3569 T^{4} + 319 p T^{5} + 30 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - T - 42 T^{2} + 31 T^{3} + 443 T^{4} + 31 p T^{5} - 42 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 7 T + 93 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T - 26 T^{2} + 32 T^{3} + 2883 T^{4} + 32 p T^{5} - 26 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + T - 74 T^{2} - 31 T^{3} + 2763 T^{4} - 31 p T^{5} - 74 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 216 T^{3} + 1119 T^{4} + 216 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T - 30 T^{2} + 304 T^{3} - 2581 T^{4} + 304 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 1691 p T^{5} + 138 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 15 T + 34 T^{2} + 675 T^{3} + 14727 T^{4} + 675 p T^{5} + 34 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 5 T^{2} + 108 T^{3} + 4584 T^{4} + 108 p T^{5} - 5 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 89 | $C_4\times C_2$ | \( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 192 p T^{5} - 50 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.30200070681610777805220320566, −7.18388788721706784508404471545, −7.06408454728152981909353254028, −6.93824675276710605621486914000, −6.58494064581342676170286637545, −5.99389102167429143345822125883, −5.81368154369663592984623651323, −5.63506866684738162765400384186, −5.58342477926546252637191606582, −5.18228379924526686173456580860, −4.89841290758881349546285374383, −4.89433108115162734843520380352, −4.47713107901044767170444803788, −4.37976855255209915232893104776, −4.07734414023770443365944455884, −3.72725999183359796478100118487, −3.61200038875644640421130128700, −2.67787970578999255388309465411, −2.48289978278159185330510584130, −2.24492335098142096033838019708, −2.06926349159960893801468188222, −1.65655772001557854254201388570, −1.49991732047244327737344784439, −1.24953647053662865152782170028, −0.22627450613446634017420649794,
0.22627450613446634017420649794, 1.24953647053662865152782170028, 1.49991732047244327737344784439, 1.65655772001557854254201388570, 2.06926349159960893801468188222, 2.24492335098142096033838019708, 2.48289978278159185330510584130, 2.67787970578999255388309465411, 3.61200038875644640421130128700, 3.72725999183359796478100118487, 4.07734414023770443365944455884, 4.37976855255209915232893104776, 4.47713107901044767170444803788, 4.89433108115162734843520380352, 4.89841290758881349546285374383, 5.18228379924526686173456580860, 5.58342477926546252637191606582, 5.63506866684738162765400384186, 5.81368154369663592984623651323, 5.99389102167429143345822125883, 6.58494064581342676170286637545, 6.93824675276710605621486914000, 7.06408454728152981909353254028, 7.18388788721706784508404471545, 7.30200070681610777805220320566
