| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s − 4·8-s + 9-s − 12·11-s − 14·13-s + 4·14-s − 12·16-s − 6·17-s + 2·18-s + 12·19-s − 24·22-s + 12·23-s − 28·26-s + 4·28-s − 2·29-s − 16·32-s − 12·34-s + 2·36-s + 8·37-s + 24·38-s − 6·41-s − 6·43-s − 24·44-s + 24·46-s − 20·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s − 1.41·8-s + 1/3·9-s − 3.61·11-s − 3.88·13-s + 1.06·14-s − 3·16-s − 1.45·17-s + 0.471·18-s + 2.75·19-s − 5.11·22-s + 2.50·23-s − 5.49·26-s + 0.755·28-s − 0.371·29-s − 2.82·32-s − 2.05·34-s + 1/3·36-s + 1.31·37-s + 3.89·38-s − 0.937·41-s − 0.914·43-s − 3.61·44-s + 3.53·46-s − 2.91·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473048844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473048844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 22 T^{3} - 68 T^{4} + 22 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 72 T^{3} + 59 T^{4} + 72 p T^{5} + 24 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 94 T^{2} - 552 T^{3} + 2667 T^{4} - 552 p T^{5} + 94 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 90 T^{2} - 504 T^{3} + 2339 T^{4} - 504 p T^{5} + 90 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 52 T^{2} - 4 T^{3} + 2179 T^{4} - 4 p T^{5} - 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 2451 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 216 T^{3} + 107 T^{4} + 216 p T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 97 T^{2} + 510 T^{3} + 5892 T^{4} + 510 p T^{5} + 97 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 240 T^{3} - 1173 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 107 T^{2} - 22 T^{3} + 8356 T^{4} - 22 p T^{5} - 107 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 121 T^{2} + 1242 T^{3} + 15012 T^{4} + 1242 p T^{5} + 121 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 144 T^{3} - 3613 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 135 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T + 184 T^{2} + 1032 T^{3} + 22731 T^{4} + 1032 p T^{5} + 184 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 26 T + 325 T^{2} - 4082 T^{3} + 48220 T^{4} - 4082 p T^{5} + 325 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.22577491817874276141009544835, −7.02172669617890531789811456920, −6.81778308920199025808024835077, −6.38492724332742250949716171348, −6.14310973732141578409462640590, −6.05727543774034306811299306411, −5.48822805131380004413032203107, −5.38576324195486280145212708975, −5.23084654268449229108372876199, −4.96778285728543956968355378161, −4.90721015185146233320028994043, −4.87711295760923943476710875680, −4.53785950985929038582079028864, −4.49240510380355209543938562531, −3.71811270813407830479229224868, −3.37776824214811318130499538891, −3.14936067752494433759763420025, −2.92109712901595911801998121848, −2.82968354386930019317707583891, −2.69303738149419749946921510041, −2.17225448219348128793023997758, −2.11469934058114748316153610013, −1.65485661416794654420197058996, −0.59413683218181612085849560278, −0.31248525208518421653984280236,
0.31248525208518421653984280236, 0.59413683218181612085849560278, 1.65485661416794654420197058996, 2.11469934058114748316153610013, 2.17225448219348128793023997758, 2.69303738149419749946921510041, 2.82968354386930019317707583891, 2.92109712901595911801998121848, 3.14936067752494433759763420025, 3.37776824214811318130499538891, 3.71811270813407830479229224868, 4.49240510380355209543938562531, 4.53785950985929038582079028864, 4.87711295760923943476710875680, 4.90721015185146233320028994043, 4.96778285728543956968355378161, 5.23084654268449229108372876199, 5.38576324195486280145212708975, 5.48822805131380004413032203107, 6.05727543774034306811299306411, 6.14310973732141578409462640590, 6.38492724332742250949716171348, 6.81778308920199025808024835077, 7.02172669617890531789811456920, 7.22577491817874276141009544835
