| L(s) = 1 | + 2·9-s − 8·11-s − 16·19-s − 4·29-s − 8·31-s − 20·41-s + 10·49-s − 4·61-s + 24·71-s − 16·79-s − 5·81-s + 12·89-s − 16·99-s − 28·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 32·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.41·11-s − 3.67·19-s − 0.742·29-s − 1.43·31-s − 3.12·41-s + 10/7·49-s − 0.512·61-s + 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s − 1.60·99-s − 2.78·101-s − 2.68·109-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.155985583096117039394650441908, −8.680928383242480127954456467440, −8.409799530723711511241889912968, −7.947159410057509323919136631618, −7.87828404512942150249527675303, −7.03480239937336302191128047163, −6.86375252968661253189757882435, −6.56907834446500933513654921958, −5.73752957257773745987183120060, −5.60104124022750292102211641871, −5.06632952221844786878583681668, −4.65346565249472860870920095398, −4.13537087465230804571820996375, −3.77112147073722392183692579348, −3.12607651914747200329058045298, −2.32443200779169547829008970144, −2.21913351617222597891132330615, −1.59187482730967088947058848206, 0, 0,
1.59187482730967088947058848206, 2.21913351617222597891132330615, 2.32443200779169547829008970144, 3.12607651914747200329058045298, 3.77112147073722392183692579348, 4.13537087465230804571820996375, 4.65346565249472860870920095398, 5.06632952221844786878583681668, 5.60104124022750292102211641871, 5.73752957257773745987183120060, 6.56907834446500933513654921958, 6.86375252968661253189757882435, 7.03480239937336302191128047163, 7.87828404512942150249527675303, 7.947159410057509323919136631618, 8.409799530723711511241889912968, 8.680928383242480127954456467440, 9.155985583096117039394650441908
