| L(s) = 1 | − 2·2-s − 3-s + 4-s + 2·6-s + 2·7-s + 2·8-s + 3·9-s − 3·11-s − 12-s + 14·13-s − 4·14-s − 4·16-s − 3·17-s − 6·18-s + 19-s − 2·21-s + 6·22-s − 9·23-s − 2·24-s + 13·25-s − 28·26-s − 8·27-s + 2·28-s − 3·29-s + 4·31-s + 2·32-s + 3·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s + 0.816·6-s + 0.755·7-s + 0.707·8-s + 9-s − 0.904·11-s − 0.288·12-s + 3.88·13-s − 1.06·14-s − 16-s − 0.727·17-s − 1.41·18-s + 0.229·19-s − 0.436·21-s + 1.27·22-s − 1.87·23-s − 0.408·24-s + 13/5·25-s − 5.49·26-s − 1.53·27-s + 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.353·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568989289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568989289\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 29 T^{2} + 8 T^{3} + 520 T^{4} + 8 p T^{5} - 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 9 T + 77 T^{2} + 450 T^{3} + 2592 T^{4} + 450 p T^{5} + 77 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T + 59 T^{2} + 168 T^{3} + 2382 T^{4} + 168 p T^{5} + 59 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 27 T + 383 T^{2} - 3780 T^{3} + 27882 T^{4} - 3780 p T^{5} + 383 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - T^{2} - 4488 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 27 T + 419 T^{2} - 4752 T^{3} + 41832 T^{4} - 4752 p T^{5} + 419 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 15 T + 101 T^{2} + 270 T^{3} - 5640 T^{4} + 270 p T^{5} + 101 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 25 T + 306 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 36750 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 132 T^{3} - 5718 T^{4} + 132 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 10 T - 86 T^{2} + 80 T^{3} + 15487 T^{4} + 80 p T^{5} - 86 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.938013502616224845201363177331, −7.911867991805647015733996704220, −7.28014965911326438153686602871, −7.13599096223703272565421893644, −7.00909713009274810126302260910, −6.45818511371480079272192802099, −6.37235885148769608984920131509, −6.29145799002460928482155558127, −5.83583070621608656820320078046, −5.75890650243240381233705541885, −5.28337723907797744907851242989, −5.05634268518439224211286119090, −5.01138476228932580053460010461, −4.43472869959853480670515477462, −4.00255217071665827496674906336, −3.99720999232823917148758877441, −3.92737892537179869299343608314, −3.44348174740318559275163512851, −3.07639802422447340172642941994, −2.35093333252676357519013681536, −2.13018686585542998227641143854, −1.98555108611611931647074163913, −1.13356926218376750695888098837, −0.896157220491492238340477353492, −0.851344135118495632976285358205,
0.851344135118495632976285358205, 0.896157220491492238340477353492, 1.13356926218376750695888098837, 1.98555108611611931647074163913, 2.13018686585542998227641143854, 2.35093333252676357519013681536, 3.07639802422447340172642941994, 3.44348174740318559275163512851, 3.92737892537179869299343608314, 3.99720999232823917148758877441, 4.00255217071665827496674906336, 4.43472869959853480670515477462, 5.01138476228932580053460010461, 5.05634268518439224211286119090, 5.28337723907797744907851242989, 5.75890650243240381233705541885, 5.83583070621608656820320078046, 6.29145799002460928482155558127, 6.37235885148769608984920131509, 6.45818511371480079272192802099, 7.00909713009274810126302260910, 7.13599096223703272565421893644, 7.28014965911326438153686602871, 7.911867991805647015733996704220, 7.938013502616224845201363177331
