| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 8·11-s − 12-s + 16-s − 8·17-s − 2·18-s − 19-s + 8·22-s − 24-s − 2·25-s + 2·27-s + 32-s − 8·33-s − 8·34-s − 2·36-s − 38-s − 4·41-s − 4·43-s + 8·44-s − 48-s + 2·49-s − 2·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 2.41·11-s − 0.288·12-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.229·19-s + 1.70·22-s − 0.204·24-s − 2/5·25-s + 0.384·27-s + 0.176·32-s − 1.39·33-s − 1.37·34-s − 1/3·36-s − 0.162·38-s − 0.624·41-s − 0.609·43-s + 1.20·44-s − 0.144·48-s + 2/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156426687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156426687\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−11.78621533917429449584146772905, −11.52980380136656255678675508354, −10.97678819419585092558064940921, −10.33021314014943111899681846904, −9.498534086647293987113947819835, −8.805995225296419929757103530190, −8.635109909037930511349667473118, −7.38500173081223226659754385270, −6.76551225484098574247524444367, −6.25428895436260465465645273423, −5.83156559761904980357839639162, −4.68758219507073195380503092750, −4.21769775255155791722086031431, −3.30499258423259072240785008617, −1.89359541125181195449265075444,
1.89359541125181195449265075444, 3.30499258423259072240785008617, 4.21769775255155791722086031431, 4.68758219507073195380503092750, 5.83156559761904980357839639162, 6.25428895436260465465645273423, 6.76551225484098574247524444367, 7.38500173081223226659754385270, 8.635109909037930511349667473118, 8.805995225296419929757103530190, 9.498534086647293987113947819835, 10.33021314014943111899681846904, 10.97678819419585092558064940921, 11.52980380136656255678675508354, 11.78621533917429449584146772905
