| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 2·15-s − 12·17-s − 8·21-s + 25-s − 4·27-s + 4·35-s + 20·43-s − 45-s − 2·49-s − 24·51-s − 12·53-s − 24·59-s + 4·61-s − 4·63-s − 4·67-s + 24·71-s + 2·75-s − 11·81-s + 12·85-s − 28·103-s + 8·105-s + 4·109-s − 12·113-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.516·15-s − 2.91·17-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 0.676·35-s + 3.04·43-s − 0.149·45-s − 2/7·49-s − 3.36·51-s − 1.64·53-s − 3.12·59-s + 0.512·61-s − 0.503·63-s − 0.488·67-s + 2.84·71-s + 0.230·75-s − 1.22·81-s + 1.30·85-s − 2.75·103-s + 0.780·105-s + 0.383·109-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.447689111417190951947467353722, −9.113263281253576510886575085659, −8.740293324354028093320034492248, −8.008294857926527437201173134092, −7.71273110823499542846308181544, −6.96167091691043492420755836355, −6.41842389830764202453884824049, −6.27087624192875571051851265258, −5.24854435453179347418331807133, −4.28530365432405873873397666950, −4.12250433368686324236236368171, −3.16807967427295861033734664069, −2.76929890617261215013507568311, −2.01836233216080453835298575952, 0,
2.01836233216080453835298575952, 2.76929890617261215013507568311, 3.16807967427295861033734664069, 4.12250433368686324236236368171, 4.28530365432405873873397666950, 5.24854435453179347418331807133, 6.27087624192875571051851265258, 6.41842389830764202453884824049, 6.96167091691043492420755836355, 7.71273110823499542846308181544, 8.008294857926527437201173134092, 8.740293324354028093320034492248, 9.113263281253576510886575085659, 9.447689111417190951947467353722
