| L(s) = 1 | − 2·2-s + 4-s + 4·7-s + 2·9-s + 2·11-s + 8·13-s − 8·14-s + 16-s − 8·17-s − 4·18-s − 4·22-s − 16·26-s + 4·28-s + 4·29-s + 2·32-s + 16·34-s + 2·36-s + 4·37-s + 12·41-s + 12·43-s + 2·44-s − 2·49-s + 8·52-s − 12·53-s − 8·58-s − 8·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s + 2/3·9-s + 0.603·11-s + 2.21·13-s − 2.13·14-s + 1/4·16-s − 1.94·17-s − 0.942·18-s − 0.852·22-s − 3.13·26-s + 0.755·28-s + 0.742·29-s + 0.353·32-s + 2.74·34-s + 1/3·36-s + 0.657·37-s + 1.87·41-s + 1.82·43-s + 0.301·44-s − 2/7·49-s + 1.10·52-s − 1.64·53-s − 1.05·58-s − 1.04·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8522848639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8522848639\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.85078764068031628780781210826, −11.34152724924583538335689542679, −11.06226320084177966638189511932, −10.76332109077944369674429873619, −10.34216843190214001821092395945, −9.364991687812248303675834307232, −9.145660341697595489858477378965, −9.030176806043449037762894238010, −8.181288028444606161175625940356, −8.079260243639446110251027998211, −7.63373519091757555317365298931, −6.55117649324971760946554101508, −6.48170268766400170628637889021, −5.78170041068762991597464841365, −4.79496409631693048558279083330, −4.32093856880512229575824594631, −3.91275786896581322341708444462, −2.68087087515998116401091200236, −1.62076057261134817791620072556, −1.05348097827727392870510429459,
1.05348097827727392870510429459, 1.62076057261134817791620072556, 2.68087087515998116401091200236, 3.91275786896581322341708444462, 4.32093856880512229575824594631, 4.79496409631693048558279083330, 5.78170041068762991597464841365, 6.48170268766400170628637889021, 6.55117649324971760946554101508, 7.63373519091757555317365298931, 8.079260243639446110251027998211, 8.181288028444606161175625940356, 9.030176806043449037762894238010, 9.145660341697595489858477378965, 9.364991687812248303675834307232, 10.34216843190214001821092395945, 10.76332109077944369674429873619, 11.06226320084177966638189511932, 11.34152724924583538335689542679, 11.85078764068031628780781210826
