L(s) = 1 | − 2·2-s + 3·3-s − 4-s + 5-s − 6·6-s − 2·7-s + 8·8-s + 2·9-s − 2·10-s − 3·11-s − 3·12-s + 9·13-s + 4·14-s + 3·15-s − 7·16-s − 2·17-s − 4·18-s + 3·19-s − 20-s − 6·21-s + 6·22-s − 23-s + 24·24-s − 8·25-s − 18·26-s − 6·27-s + 2·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.73·3-s − 1/2·4-s + 0.447·5-s − 2.44·6-s − 0.755·7-s + 2.82·8-s + 2/3·9-s − 0.632·10-s − 0.904·11-s − 0.866·12-s + 2.49·13-s + 1.06·14-s + 0.774·15-s − 7/4·16-s − 0.485·17-s − 0.942·18-s + 0.688·19-s − 0.223·20-s − 1.30·21-s + 1.27·22-s − 0.208·23-s + 4.89·24-s − 8/5·25-s − 3.53·26-s − 1.15·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36493681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36493681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 863 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 19 T + 207 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 21 T + 233 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 25 T + 313 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 185 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−7.966164147079045004430370908122, −7.903593883955323280920483257250, −7.42825530040303276910612644853, −7.24082249208048815642605076363, −6.54443210285323835346686876076, −6.19755158783371892609049136812, −5.61967847425132467550273266263, −5.60317516235288202814959805631, −5.04506575913172768637878753450, −4.48049639839874853547067916470, −3.92688796547235443293728004325, −3.80619497765751653309732269049, −3.29277313806043495190102038987, −3.21592732367449984372972918221, −2.29510876803931096661565979706, −2.12493957335690480931053399602, −1.37755701035714148138910374502, −1.22875976686850573077630225103, 0, 0,
1.22875976686850573077630225103, 1.37755701035714148138910374502, 2.12493957335690480931053399602, 2.29510876803931096661565979706, 3.21592732367449984372972918221, 3.29277313806043495190102038987, 3.80619497765751653309732269049, 3.92688796547235443293728004325, 4.48049639839874853547067916470, 5.04506575913172768637878753450, 5.60317516235288202814959805631, 5.61967847425132467550273266263, 6.19755158783371892609049136812, 6.54443210285323835346686876076, 7.24082249208048815642605076363, 7.42825530040303276910612644853, 7.903593883955323280920483257250, 7.966164147079045004430370908122