| L(s) = 1 | + 2·2-s − 4·3-s + 2·4-s − 8·6-s − 4·8-s + 8·9-s − 12·11-s − 8·12-s + 6·13-s − 12·16-s + 6·17-s + 16·18-s − 24·22-s + 18·23-s + 16·24-s + 12·26-s − 8·27-s − 16·32-s + 48·33-s + 12·34-s + 16·36-s − 12·37-s − 24·39-s + 36·41-s + 18·43-s − 24·44-s + 36·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 4-s − 3.26·6-s − 1.41·8-s + 8/3·9-s − 3.61·11-s − 2.30·12-s + 1.66·13-s − 3·16-s + 1.45·17-s + 3.77·18-s − 5.11·22-s + 3.75·23-s + 3.26·24-s + 2.35·26-s − 1.53·27-s − 2.82·32-s + 8.35·33-s + 2.05·34-s + 8/3·36-s − 1.97·37-s − 3.84·39-s + 5.62·41-s + 2.74·43-s − 3.61·44-s + 5.30·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9690508734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9690508734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} + 48 T^{3} - 313 T^{4} + 48 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 49 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 4871 T^{4} - 972 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 31 T^{2} + 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 18 T + 149 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 5711 T^{4} - 972 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 1609 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 392 T^{3} - 3593 T^{4} - 392 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 18 T + 162 T^{2} + 504 T^{3} + 47 T^{4} + 504 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 27 T + 314 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 5471 T^{4} - 972 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^3$ | \( 1 - 13522 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 151 T^{2} + 14880 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 18 T + 162 T^{2} - 576 T^{3} - 14593 T^{4} - 576 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.69540992064172830868948423307, −7.66395905494722525891822095458, −7.29286557286104022865145250787, −7.12736506268770695545830710877, −7.12371683678637675777380330969, −6.40480796255843677033044272529, −6.12684425510136513422350818765, −6.05640121382619513815959987789, −5.89233348866537926980084131478, −5.72463567956214512445298241264, −5.40433421362958213931506952698, −5.32610586529493441473896479228, −5.18640491165437256588786643185, −4.69686469928771239085555857627, −4.46009801485660339381640929699, −4.41168364659934296721535674079, −3.72514939022286063653337631210, −3.57267483935126400575599619429, −2.96732553393048540926660150461, −2.71517841231273306693443464226, −2.70449167663956112806736102744, −2.62579085897503827920272131754, −1.31511007440766746168914720358, −1.02971206512541623959584392745, −0.38369220967169897146009980043,
0.38369220967169897146009980043, 1.02971206512541623959584392745, 1.31511007440766746168914720358, 2.62579085897503827920272131754, 2.70449167663956112806736102744, 2.71517841231273306693443464226, 2.96732553393048540926660150461, 3.57267483935126400575599619429, 3.72514939022286063653337631210, 4.41168364659934296721535674079, 4.46009801485660339381640929699, 4.69686469928771239085555857627, 5.18640491165437256588786643185, 5.32610586529493441473896479228, 5.40433421362958213931506952698, 5.72463567956214512445298241264, 5.89233348866537926980084131478, 6.05640121382619513815959987789, 6.12684425510136513422350818765, 6.40480796255843677033044272529, 7.12371683678637675777380330969, 7.12736506268770695545830710877, 7.29286557286104022865145250787, 7.66395905494722525891822095458, 7.69540992064172830868948423307
