| L(s) = 1 | + 3-s − 4·5-s + 9-s − 4·15-s + 8·19-s + 16·23-s + 2·25-s + 27-s + 12·29-s − 8·43-s − 4·45-s − 14·49-s − 4·53-s + 8·57-s + 8·67-s + 16·69-s − 16·71-s + 20·73-s + 2·75-s + 81-s + 12·87-s − 32·95-s + 4·97-s − 36·101-s − 64·115-s − 6·121-s + 28·125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.03·15-s + 1.83·19-s + 3.33·23-s + 2/5·25-s + 0.192·27-s + 2.22·29-s − 1.21·43-s − 0.596·45-s − 2·49-s − 0.549·53-s + 1.05·57-s + 0.977·67-s + 1.92·69-s − 1.89·71-s + 2.34·73-s + 0.230·75-s + 1/9·81-s + 1.28·87-s − 3.28·95-s + 0.406·97-s − 3.58·101-s − 5.96·115-s − 0.545·121-s + 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160141318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160141318\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.92388151188979644343896213523, −9.780091138821383013974264992430, −9.656797755147918818405236990829, −8.884556993307409273860919130264, −8.173282349834764667604075089133, −8.157389656224392853374347080198, −7.26958585488936859387169650645, −7.06857822018944724974086155331, −6.38250152569089720234352942221, −5.01357758335939619070346679818, −4.99175853408267534988469991920, −4.02227786472289347756607333391, −3.14826068230388029805668446658, −3.04433904291477058087427796742, −1.12835632773744922111480113749,
1.12835632773744922111480113749, 3.04433904291477058087427796742, 3.14826068230388029805668446658, 4.02227786472289347756607333391, 4.99175853408267534988469991920, 5.01357758335939619070346679818, 6.38250152569089720234352942221, 7.06857822018944724974086155331, 7.26958585488936859387169650645, 8.157389656224392853374347080198, 8.173282349834764667604075089133, 8.884556993307409273860919130264, 9.656797755147918818405236990829, 9.780091138821383013974264992430, 10.92388151188979644343896213523
