| L(s) = 1 | + 2·2-s + 4·3-s + 8·6-s − 2·8-s + 10·9-s − 2·13-s − 2·16-s − 2·17-s + 20·18-s + 12·19-s + 10·23-s − 8·24-s − 4·26-s + 20·27-s − 6·29-s + 8·31-s − 4·32-s − 4·34-s + 24·37-s + 24·38-s − 8·39-s − 4·41-s + 8·43-s + 20·46-s + 10·47-s − 8·48-s − 8·51-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 3.26·6-s − 0.707·8-s + 10/3·9-s − 0.554·13-s − 1/2·16-s − 0.485·17-s + 4.71·18-s + 2.75·19-s + 2.08·23-s − 1.63·24-s − 0.784·26-s + 3.84·27-s − 1.11·29-s + 1.43·31-s − 0.707·32-s − 0.685·34-s + 3.94·37-s + 3.89·38-s − 1.28·39-s − 0.624·41-s + 1.21·43-s + 2.94·46-s + 1.45·47-s − 1.15·48-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(46.86858525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(46.86858525\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 26 T^{2} - 14 T^{3} + 360 T^{4} - 14 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 24 T^{2} + 42 T^{3} + 413 T^{4} + 42 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 40 T^{2} + 80 T^{3} + 916 T^{4} + 80 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 12 T + 6 p T^{2} - 724 T^{3} + 3619 T^{4} - 724 p T^{5} + 6 p^{3} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 10 T + 60 T^{2} - 104 T^{3} + 196 T^{4} - 104 p T^{5} + 60 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 332 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 8 T + 118 T^{2} - 648 T^{3} + 5455 T^{4} - 648 p T^{5} + 118 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 24 T + 352 T^{2} - 3386 T^{3} + 24217 T^{4} - 3386 p T^{5} + 352 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 114 T^{2} + 346 T^{3} + 5976 T^{4} + 346 p T^{5} + 114 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 140 T^{2} - 878 T^{3} + 8293 T^{4} - 878 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 10 T + 204 T^{2} - 1346 T^{3} + 14638 T^{4} - 1346 p T^{5} + 204 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 20 T + 332 T^{2} - 3388 T^{3} + 29670 T^{4} - 3388 p T^{5} + 332 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 2 T + 230 T^{2} + 348 T^{3} + 20188 T^{4} + 348 p T^{5} + 230 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 8 T + 144 T^{2} - 764 T^{3} + 10626 T^{4} - 764 p T^{5} + 144 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 196 T^{2} - 662 T^{3} + 16381 T^{4} - 662 p T^{5} + 196 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 1648 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 12 T + 228 T^{2} + 1834 T^{3} + 21241 T^{4} + 1834 p T^{5} + 228 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8 T + 162 T^{2} + 1212 T^{3} + 20195 T^{4} + 1212 p T^{5} + 162 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 6 T + 276 T^{2} - 1256 T^{3} + 32400 T^{4} - 1256 p T^{5} + 276 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 162 T^{2} + 1150 T^{3} + 140 p T^{4} + 1150 p T^{5} + 162 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 2 T + 380 T^{2} - 566 T^{3} + 54898 T^{4} - 566 p T^{5} + 380 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.07115634668205915301457593175, −5.55744163884573710015568280904, −5.45919957393298610186249379859, −5.35831839156564903470419855864, −5.34640805166798667665358572527, −4.92034049905551311549512780546, −4.65875128831286096060760721477, −4.45315122789529564647575889825, −4.45041491768787536258457177940, −4.06578508198054488877092597398, −4.01622250719602074733974298218, −4.00428740992840496967808074190, −3.64295987000612534115901812462, −3.22002901527268817969185850914, −3.00700869999384132679884919248, −2.92293764554115360352850119766, −2.79541531124375583619443519623, −2.68897157475650331519657934350, −2.34893930314167339552875072365, −2.01922528300175350015662260436, −1.79603314811612369665355365591, −1.34327876640969372456032338398, −1.03449169426837088275268248027, −0.72074721297206983525552566513, −0.70789191304017678213872932895,
0.70789191304017678213872932895, 0.72074721297206983525552566513, 1.03449169426837088275268248027, 1.34327876640969372456032338398, 1.79603314811612369665355365591, 2.01922528300175350015662260436, 2.34893930314167339552875072365, 2.68897157475650331519657934350, 2.79541531124375583619443519623, 2.92293764554115360352850119766, 3.00700869999384132679884919248, 3.22002901527268817969185850914, 3.64295987000612534115901812462, 4.00428740992840496967808074190, 4.01622250719602074733974298218, 4.06578508198054488877092597398, 4.45041491768787536258457177940, 4.45315122789529564647575889825, 4.65875128831286096060760721477, 4.92034049905551311549512780546, 5.34640805166798667665358572527, 5.35831839156564903470419855864, 5.45919957393298610186249379859, 5.55744163884573710015568280904, 6.07115634668205915301457593175
