| L(s) = 1 | + 9-s + 6·11-s + 3·17-s − 5·19-s − 2·25-s − 19·41-s + 7·43-s + 10·49-s + 10·59-s − 7·67-s − 26·73-s + 81-s − 15·83-s + 3·89-s − 20·97-s + 6·99-s + 16·107-s − 30·113-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.80·11-s + 0.727·17-s − 1.14·19-s − 2/5·25-s − 2.96·41-s + 1.06·43-s + 10/7·49-s + 1.30·59-s − 0.855·67-s − 3.04·73-s + 1/9·81-s − 1.64·83-s + 0.317·89-s − 2.03·97-s + 0.603·99-s + 1.54·107-s − 2.82·113-s + 9/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.242·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−7.26695939617069538876067925638, −6.97073969142367994473625678713, −6.67187991943325321193145214147, −6.12901251037511005053083158556, −5.80817403433573968792645857563, −5.35823734083993733893278257968, −4.74073771014528297827263498481, −4.21453779075228920835260968405, −3.97858078884245832106378856264, −3.53493743192059232851524396445, −2.90019715749081364049769555859, −2.26635840593105874802629410681, −1.51145688633431541605080571453, −1.23988828577819986809636614169, 0,
1.23988828577819986809636614169, 1.51145688633431541605080571453, 2.26635840593105874802629410681, 2.90019715749081364049769555859, 3.53493743192059232851524396445, 3.97858078884245832106378856264, 4.21453779075228920835260968405, 4.74073771014528297827263498481, 5.35823734083993733893278257968, 5.80817403433573968792645857563, 6.12901251037511005053083158556, 6.67187991943325321193145214147, 6.97073969142367994473625678713, 7.26695939617069538876067925638
