| L(s) = 1 | − 2·5-s + 9-s + 12·23-s − 25-s + 4·31-s − 4·37-s − 2·45-s − 6·49-s − 8·53-s − 4·67-s + 8·71-s + 81-s − 24·89-s − 28·97-s + 28·103-s − 24·113-s − 24·115-s − 11·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·155-s + 157-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s + 2.50·23-s − 1/5·25-s + 0.718·31-s − 0.657·37-s − 0.298·45-s − 6/7·49-s − 1.09·53-s − 0.488·67-s + 0.949·71-s + 1/9·81-s − 2.54·89-s − 2.84·97-s + 2.75·103-s − 2.25·113-s − 2.23·115-s − 121-s + 1.07·125-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 − 0.642·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.69115118923176121385716738938, −7.17883929065172577157306747827, −6.75061686203910772180860846224, −6.57629659116561447150976171988, −5.87982693700294744230920296506, −5.31353892677649519944522858113, −4.95056378158698007774502792390, −4.53892176567725626873733705917, −4.01515344282807457573055846116, −3.53397017310041728257276381158, −2.98755384338560128265718945237, −2.61401214957347435950459893089, −1.61590889121649202500726256173, −1.06216627539198540865057142025, 0,
1.06216627539198540865057142025, 1.61590889121649202500726256173, 2.61401214957347435950459893089, 2.98755384338560128265718945237, 3.53397017310041728257276381158, 4.01515344282807457573055846116, 4.53892176567725626873733705917, 4.95056378158698007774502792390, 5.31353892677649519944522858113, 5.87982693700294744230920296506, 6.57629659116561447150976171988, 6.75061686203910772180860846224, 7.17883929065172577157306747827, 7.69115118923176121385716738938
