| L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s + 16-s − 8·17-s − 2·18-s − 4·23-s − 25-s − 4·31-s − 32-s + 8·34-s + 2·36-s − 4·41-s + 4·46-s + 49-s + 50-s + 4·62-s + 64-s − 8·68-s − 12·71-s − 2·72-s + 12·79-s − 5·81-s + 4·82-s − 16·89-s − 4·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.834·23-s − 1/5·25-s − 0.718·31-s − 0.176·32-s + 1.37·34-s + 1/3·36-s − 0.624·41-s + 0.589·46-s + 1/7·49-s + 0.141·50-s + 0.508·62-s + 1/8·64-s − 0.970·68-s − 1.42·71-s − 0.235·72-s + 1.35·79-s − 5/9·81-s + 0.441·82-s − 1.69·89-s − 0.417·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.867455662518271704474267937595, −8.713605111377536793193249020948, −8.100846656661504319534059098960, −7.58312910542297095523315876583, −7.07415746578698148349393407764, −6.69537954885867781037502984153, −6.20290133401924309888325379104, −5.59418541797818041055403410070, −4.91718390234939527067783290265, −4.23757328662087924083042675448, −3.86885070680044093783100576049, −2.89184784540670637393772956941, −2.16659213165717811168901441805, −1.51462205340742464024397314015, 0,
1.51462205340742464024397314015, 2.16659213165717811168901441805, 2.89184784540670637393772956941, 3.86885070680044093783100576049, 4.23757328662087924083042675448, 4.91718390234939527067783290265, 5.59418541797818041055403410070, 6.20290133401924309888325379104, 6.69537954885867781037502984153, 7.07415746578698148349393407764, 7.58312910542297095523315876583, 8.100846656661504319534059098960, 8.713605111377536793193249020948, 8.867455662518271704474267937595
