| 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
| \(L(1)\) |
\(\approx\) |
\(4.873176677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.873176677\) |
| \(L(\frac{3}{2})\) |
|
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
−7.17052833326586078246419207039, −6.84756478257001920402433942551, −6.77511062960964772944496792204, −6.70825218354503709394734013670, −6.53589707938970740461745120739, −5.96390155268197329292993196691, −5.92301198136373501534344848393, −5.88701186341722526007489620281, −5.77679642230245951830913397835, −5.27904261213370134665554667892, −5.00953244460044038395992116325, −4.84229276033323411251039063023, −4.36204686095766813351087691247, −4.08380957768712599306680218942, −3.75987645912642624103158092316, −3.52226367798624950678381557565, −3.43369735316882187299810339448, −3.40005249612749518860833855928, −2.80179031813826755307287236465, −2.64353631469007009791230489217, −2.37231000877518745962369050678, −1.54432351847214216908290375232, −1.16697547126976525393050231525, −1.12699977150721746510438309398, −0.61474038380761776604481348661,
0.61474038380761776604481348661, 1.12699977150721746510438309398, 1.16697547126976525393050231525, 1.54432351847214216908290375232, 2.37231000877518745962369050678, 2.64353631469007009791230489217, 2.80179031813826755307287236465, 3.40005249612749518860833855928, 3.43369735316882187299810339448, 3.52226367798624950678381557565, 3.75987645912642624103158092316, 4.08380957768712599306680218942, 4.36204686095766813351087691247, 4.84229276033323411251039063023, 5.00953244460044038395992116325, 5.27904261213370134665554667892, 5.77679642230245951830913397835, 5.88701186341722526007489620281, 5.92301198136373501534344848393, 5.96390155268197329292993196691, 6.53589707938970740461745120739, 6.70825218354503709394734013670, 6.77511062960964772944496792204, 6.84756478257001920402433942551, 7.17052833326586078246419207039
