| L(s) = 1 | + 2·2-s + 2·4-s − 2·11-s + 10·13-s − 4·16-s − 4·22-s + 10·23-s − 6·25-s + 20·26-s − 8·32-s + 4·37-s − 4·44-s + 20·46-s − 20·47-s + 10·49-s − 12·50-s + 20·52-s + 10·59-s − 8·64-s + 10·73-s + 8·74-s + 8·83-s + 20·92-s − 40·94-s + 16·97-s + 20·98-s − 12·100-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.603·11-s + 2.77·13-s − 16-s − 0.852·22-s + 2.08·23-s − 6/5·25-s + 3.92·26-s − 1.41·32-s + 0.657·37-s − 0.603·44-s + 2.94·46-s − 2.91·47-s + 10/7·49-s − 1.69·50-s + 2.77·52-s + 1.30·59-s − 64-s + 1.17·73-s + 0.929·74-s + 0.878·83-s + 2.08·92-s − 4.12·94-s + 1.62·97-s + 2.02·98-s − 6/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.649847049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.649847049\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.202469587591768667924272719031, −8.717308396966447806996876277268, −8.298941545720346037958699029104, −7.83837065900924055390337470659, −7.05411705082100059481195684685, −6.58763688589681282982455484519, −6.16255785550171488591192676722, −5.72053532078692254034988768397, −5.18126023985763114483401436137, −4.71050583168061410079956226001, −3.91595087630796138809517480082, −3.56562195258391274536449768199, −3.07371919967314823514888267675, −2.19529546575372813775039234544, −1.13134333929750020751305195928,
1.13134333929750020751305195928, 2.19529546575372813775039234544, 3.07371919967314823514888267675, 3.56562195258391274536449768199, 3.91595087630796138809517480082, 4.71050583168061410079956226001, 5.18126023985763114483401436137, 5.72053532078692254034988768397, 6.16255785550171488591192676722, 6.58763688589681282982455484519, 7.05411705082100059481195684685, 7.83837065900924055390337470659, 8.298941545720346037958699029104, 8.717308396966447806996876277268, 9.202469587591768667924272719031
