| L(s) = 1 | − 6·5-s + 9-s − 10·13-s + 2·17-s + 17·25-s + 4·29-s + 16·37-s + 14·41-s − 6·45-s + 2·49-s − 4·53-s + 16·61-s + 60·65-s − 28·73-s + 81-s − 12·85-s + 16·89-s + 24·97-s − 40·109-s − 38·113-s − 10·117-s − 21·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1/3·9-s − 2.77·13-s + 0.485·17-s + 17/5·25-s + 0.742·29-s + 2.63·37-s + 2.18·41-s − 0.894·45-s + 2/7·49-s − 0.549·53-s + 2.04·61-s + 7.44·65-s − 3.27·73-s + 1/9·81-s − 1.30·85-s + 1.69·89-s + 2.43·97-s − 3.83·109-s − 3.57·113-s − 0.924·117-s − 1.90·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−8.164472493503383013355655659064, −7.928512887641711908544810696892, −7.72417736726020375123060781448, −7.17889147799285411822854898551, −7.10862606316740176491095490628, −6.25936254419965766006908676624, −5.57392449844244671802827644030, −4.81536414408362940263533105554, −4.54778361443587554242160787221, −4.13220745200772588524548445751, −3.66741791988930080526156695167, −2.63789286879136207172642804519, −2.63752616145671866471612418575, −0.906565211409185083613124704727, 0,
0.906565211409185083613124704727, 2.63752616145671866471612418575, 2.63789286879136207172642804519, 3.66741791988930080526156695167, 4.13220745200772588524548445751, 4.54778361443587554242160787221, 4.81536414408362940263533105554, 5.57392449844244671802827644030, 6.25936254419965766006908676624, 7.10862606316740176491095490628, 7.17889147799285411822854898551, 7.72417736726020375123060781448, 7.928512887641711908544810696892, 8.164472493503383013355655659064
