| L(s) = 1 | + 8·5-s + 18·9-s + 48·13-s + 39·25-s − 48·37-s + 36·41-s + 144·45-s − 98·49-s + 112·53-s + 384·65-s + 243·81-s + 156·89-s + 864·117-s − 242·121-s + 112·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.39e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 2·9-s + 3.69·13-s + 1.55·25-s − 1.29·37-s + 0.878·41-s + 16/5·45-s − 2·49-s + 2.11·53-s + 5.90·65-s + 3·81-s + 1.75·89-s + 7.38·117-s − 2·121-s + 0.895·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 8.22·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.952244255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.952244255\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 8 T + p^{2} T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )( 1 + 120 T + p^{2} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
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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
−9.607041570317001048989775741099, −9.331551270187722026533926287729, −8.802053450013597114337777975552, −8.758619449428116650852148281949, −8.000624013556023584466850056336, −7.81083193487931465010120954474, −6.89345932991980956682527666917, −6.81643088156873372417618589093, −6.26608820256039565067971613378, −6.15218148439034175071161653978, −5.42671784189869688735405465885, −5.30518183377322114864348104245, −4.28646744912805285382355175469, −4.25508132270218301173856621775, −3.42303031969830355276022594145, −3.33686222835430886248739886247, −2.14337913506181566046259105284, −1.81534527348050010406486854434, −1.13926489705976972608116953224, −1.05703544184845788255551965757,
1.05703544184845788255551965757, 1.13926489705976972608116953224, 1.81534527348050010406486854434, 2.14337913506181566046259105284, 3.33686222835430886248739886247, 3.42303031969830355276022594145, 4.25508132270218301173856621775, 4.28646744912805285382355175469, 5.30518183377322114864348104245, 5.42671784189869688735405465885, 6.15218148439034175071161653978, 6.26608820256039565067971613378, 6.81643088156873372417618589093, 6.89345932991980956682527666917, 7.81083193487931465010120954474, 8.000624013556023584466850056336, 8.758619449428116650852148281949, 8.802053450013597114337777975552, 9.331551270187722026533926287729, 9.607041570317001048989775741099
