| L(s) = 1 | + 2·3-s − 4·7-s + 9-s − 4·13-s − 8·19-s − 8·21-s − 4·27-s − 8·31-s − 4·37-s − 8·39-s + 20·43-s − 2·49-s − 16·57-s + 4·61-s − 4·63-s − 4·67-s − 4·73-s + 16·79-s − 11·81-s + 16·91-s − 16·93-s − 4·97-s − 28·103-s + 4·109-s − 8·111-s − 4·117-s − 22·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 1.74·21-s − 0.769·27-s − 1.43·31-s − 0.657·37-s − 1.28·39-s + 3.04·43-s − 2/7·49-s − 2.11·57-s + 0.512·61-s − 0.503·63-s − 0.488·67-s − 0.468·73-s + 1.80·79-s − 1.22·81-s + 1.67·91-s − 1.65·93-s − 0.406·97-s − 2.75·103-s + 0.383·109-s − 0.759·111-s − 0.369·117-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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, −8.414042470938792927705837581638, −8.008294857926527437201173134092, −7.38409000222510611898226618037, −6.96167091691043492420755836355, −6.41842389830764202453884824049, −5.85740317846722582719715899176, −5.24854435453179347418331807133, −4.28530365432405873873397666950, −3.90074996505307731038192172560, −3.16807967427295861033734664069, −2.58321256178540655780606724063, −2.01836233216080453835298575952, 0,
2.01836233216080453835298575952, 2.58321256178540655780606724063, 3.16807967427295861033734664069, 3.90074996505307731038192172560, 4.28530365432405873873397666950, 5.24854435453179347418331807133, 5.85740317846722582719715899176, 6.41842389830764202453884824049, 6.96167091691043492420755836355, 7.38409000222510611898226618037, 8.008294857926527437201173134092, 8.414042470938792927705837581638, 9.113263281253576510886575085659, 9.447689111417190951947467353722
