| L(s) = 1 | + 4·5-s − 4·11-s − 16·19-s + 11·25-s − 16·29-s − 4·41-s + 10·49-s − 16·55-s − 12·59-s − 4·61-s − 8·71-s − 16·79-s + 12·89-s − 64·95-s − 12·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s − 3.67·19-s + 11/5·25-s − 2.97·29-s − 0.624·41-s + 10/7·49-s − 2.15·55-s − 1.56·59-s − 0.512·61-s − 0.949·71-s − 1.80·79-s + 1.27·89-s − 6.56·95-s − 1.14·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9049976190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9049976190\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.268177394708323029039873049736, −8.587390452096719113575489504736, −8.423832746211050667883485286644, −7.74239352074784026533106070803, −7.54580330778737615263910704726, −6.91979212716922612287566611429, −6.69090216278290355321555768848, −6.09704490451645297475332832255, −5.98311954980599373297149254192, −5.57009367277100944341182535334, −5.28216964821954324955788193451, −4.59781379018489184756234385581, −4.41696931555720389855660790343, −3.84826378780299570183105465452, −3.27041264652413628877795841759, −2.46447590113798284131579703632, −2.46382969019994924043429099589, −1.71037523096834087919420244179, −1.69861728057308960598431550052, −0.26910078834385321988759637790,
0.26910078834385321988759637790, 1.69861728057308960598431550052, 1.71037523096834087919420244179, 2.46382969019994924043429099589, 2.46447590113798284131579703632, 3.27041264652413628877795841759, 3.84826378780299570183105465452, 4.41696931555720389855660790343, 4.59781379018489184756234385581, 5.28216964821954324955788193451, 5.57009367277100944341182535334, 5.98311954980599373297149254192, 6.09704490451645297475332832255, 6.69090216278290355321555768848, 6.91979212716922612287566611429, 7.54580330778737615263910704726, 7.74239352074784026533106070803, 8.423832746211050667883485286644, 8.587390452096719113575489504736, 9.268177394708323029039873049736
