| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 2·8-s − 8·9-s + 4·10-s + 8·11-s + 14·13-s − 6·17-s − 16·18-s − 24·19-s + 6·20-s + 16·22-s + 3·25-s + 28·26-s − 14·29-s + 8·31-s − 6·32-s − 12·34-s − 24·36-s + 2·37-s − 48·38-s + 4·40-s + 6·41-s − 4·43-s + 24·44-s − 16·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.707·8-s − 8/3·9-s + 1.26·10-s + 2.41·11-s + 3.88·13-s − 1.45·17-s − 3.77·18-s − 5.50·19-s + 1.34·20-s + 3.41·22-s + 3/5·25-s + 5.49·26-s − 2.59·29-s + 1.43·31-s − 1.06·32-s − 2.05·34-s − 4·36-s + 0.328·37-s − 7.78·38-s + 0.632·40-s + 0.937·41-s − 0.609·43-s + 3.61·44-s − 2.38·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.502986169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.502986169\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 14 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} + 6 T^{3} + 324 T^{4} + 6 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 44 T^{2} + 1407 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 574 T^{3} + 3276 T^{4} + 574 p T^{5} + 97 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 12 T^{2} - 112 T^{3} + 2831 T^{4} - 112 p T^{5} - 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 72 T^{2} + 8 T^{3} + 5207 T^{4} + 8 p T^{5} - 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 50 T^{2} + 112 T^{3} + 1395 T^{4} + 112 p T^{5} - 50 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T - 2 T^{2} - 48 T^{3} + 6051 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 10 T - 39 T^{2} - 70 T^{3} + 6692 T^{4} - 70 p T^{5} - 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T - 32 T^{2} + 216 T^{3} + 13359 T^{4} + 216 p T^{5} - 32 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} + 896 T^{3} - 9261 T^{4} + 896 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 6 T^{2} + 896 T^{3} + 28259 T^{4} + 896 p T^{5} + 6 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.84200013322653841306014851527, −7.07796977187795315354882140371, −6.77920429459994207818461666072, −6.65102137448723698056869830160, −6.39573170348519056064965507208, −6.34395612266648988756267522015, −6.10602012087559871308793263449, −6.07793391307184304977523455296, −5.92426066861154398834585733432, −5.57215493612660347113150466940, −5.49770218290396218485595032626, −4.83803270322611620056216274970, −4.43074281108618830760897749975, −4.26911879268497654863850389238, −4.21209000816355882694270484298, −3.88240260890920494788330819234, −3.64390134104786240496905395485, −3.42068674476452744574649696285, −2.96024514326622300271566567045, −2.60290540300076696219005675464, −2.47732667718206220225326324132, −1.78750080670595596117854158120, −1.75981270760170625448789626429, −1.50730370492035605962980105886, −0.36432858197826797794647474609,
0.36432858197826797794647474609, 1.50730370492035605962980105886, 1.75981270760170625448789626429, 1.78750080670595596117854158120, 2.47732667718206220225326324132, 2.60290540300076696219005675464, 2.96024514326622300271566567045, 3.42068674476452744574649696285, 3.64390134104786240496905395485, 3.88240260890920494788330819234, 4.21209000816355882694270484298, 4.26911879268497654863850389238, 4.43074281108618830760897749975, 4.83803270322611620056216274970, 5.49770218290396218485595032626, 5.57215493612660347113150466940, 5.92426066861154398834585733432, 6.07793391307184304977523455296, 6.10602012087559871308793263449, 6.34395612266648988756267522015, 6.39573170348519056064965507208, 6.65102137448723698056869830160, 6.77920429459994207818461666072, 7.07796977187795315354882140371, 7.84200013322653841306014851527
