| L(s) = 1 | + 6·5-s + 9-s + 6·13-s − 2·17-s + 17·25-s + 12·29-s − 8·37-s + 18·41-s + 6·45-s − 14·49-s − 4·53-s − 24·61-s + 36·65-s − 20·73-s + 81-s − 12·85-s + 8·89-s + 24·97-s − 16·101-s − 24·109-s + 6·113-s + 6·117-s − 21·121-s + 18·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 1/3·9-s + 1.66·13-s − 0.485·17-s + 17/5·25-s + 2.22·29-s − 1.31·37-s + 2.81·41-s + 0.894·45-s − 2·49-s − 0.549·53-s − 3.07·61-s + 4.46·65-s − 2.34·73-s + 1/9·81-s − 1.30·85-s + 0.847·89-s + 2.43·97-s − 1.59·101-s − 2.29·109-s + 0.564·113-s + 0.554·117-s − 1.90·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.571129383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.571129383\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−8.730686790656812538651452890426, −8.637357193637508785862031083481, −7.85592526256654437336635861466, −7.32843269717286178163258400168, −6.49555364164429910198281440649, −6.29479205975676746085811661026, −6.09600151641730469216732038367, −5.62818700463789751758773761734, −4.87102240857422416675915309029, −4.59174724472301644959370410844, −3.72825125753065481759290584299, −2.89883589273214610624179644709, −2.48880099807193300568441320852, −1.50390211211732167141396345113, −1.39385455464072664196875612123,
1.39385455464072664196875612123, 1.50390211211732167141396345113, 2.48880099807193300568441320852, 2.89883589273214610624179644709, 3.72825125753065481759290584299, 4.59174724472301644959370410844, 4.87102240857422416675915309029, 5.62818700463789751758773761734, 6.09600151641730469216732038367, 6.29479205975676746085811661026, 6.49555364164429910198281440649, 7.32843269717286178163258400168, 7.85592526256654437336635861466, 8.637357193637508785862031083481, 8.730686790656812538651452890426
