| L(s) = 1 | + 2·2-s + 3·3-s + 2·4-s + 6·6-s + 6·9-s + 4·11-s + 6·12-s − 5·13-s − 4·16-s + 12·18-s + 8·22-s + 2·23-s + 4·25-s − 10·26-s + 9·27-s − 8·32-s + 12·33-s + 12·36-s − 3·37-s − 15·39-s + 8·44-s + 4·46-s − 2·47-s − 12·48-s + 8·49-s + 8·50-s − 10·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 4-s + 2.44·6-s + 2·9-s + 1.20·11-s + 1.73·12-s − 1.38·13-s − 16-s + 2.82·18-s + 1.70·22-s + 0.417·23-s + 4/5·25-s − 1.96·26-s + 1.73·27-s − 1.41·32-s + 2.08·33-s + 2·36-s − 0.493·37-s − 2.40·39-s + 1.20·44-s + 0.589·46-s − 0.291·47-s − 1.73·48-s + 8/7·49-s + 1.13·50-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.180658969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.180658969\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
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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
−9.189205955243618656277902619704, −8.818173684743094020641390290529, −8.410405982392944812001119018277, −7.73379308477398937245779421947, −7.13017704584995261759014575507, −6.96445833066424659575867986460, −6.36039266372128574947182225921, −5.64471062059894606116681232324, −4.89486348860187773780886657864, −4.62032131265335611920254707620, −3.84658568146228851118138813514, −3.60919842075546499506352529540, −2.75553336361346751146598471942, −2.48929243388103574004776669545, −1.51623013343341193389598043538,
1.51623013343341193389598043538, 2.48929243388103574004776669545, 2.75553336361346751146598471942, 3.60919842075546499506352529540, 3.84658568146228851118138813514, 4.62032131265335611920254707620, 4.89486348860187773780886657864, 5.64471062059894606116681232324, 6.36039266372128574947182225921, 6.96445833066424659575867986460, 7.13017704584995261759014575507, 7.73379308477398937245779421947, 8.410405982392944812001119018277, 8.818173684743094020641390290529, 9.189205955243618656277902619704
