| L(s) = 1 | + 2·3-s + 4-s − 6·5-s + 3·9-s + 6·11-s + 2·12-s − 12·15-s + 16-s − 6·20-s + 17·25-s + 4·27-s − 8·31-s + 12·33-s + 3·36-s − 2·37-s + 6·44-s − 18·45-s − 6·47-s + 2·48-s + 11·49-s − 12·53-s − 36·55-s + 6·59-s − 12·60-s + 64-s − 8·67-s + 24·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 2.68·5-s + 9-s + 1.80·11-s + 0.577·12-s − 3.09·15-s + 1/4·16-s − 1.34·20-s + 17/5·25-s + 0.769·27-s − 1.43·31-s + 2.08·33-s + 1/2·36-s − 0.328·37-s + 0.904·44-s − 2.68·45-s − 0.875·47-s + 0.288·48-s + 11/7·49-s − 1.64·53-s − 4.85·55-s + 0.781·59-s − 1.54·60-s + 1/8·64-s − 0.977·67-s + 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3663396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3663396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.27784894919030119363644743187, −6.97231192698558333316844758012, −6.79661597729253634860739013116, −6.20094236439109133333596148959, −5.62068913980564779903495376901, −4.85512283254279582558561488230, −4.54545014782265603366854791153, −3.88782305896589818564941135966, −3.70435541571971822099748167322, −3.64826604965760821530726335042, −2.99941437940098712138765854096, −2.34583244545989611355087103303, −1.63272357936029339660940403376, −1.01703241328653437689861552785, 0,
1.01703241328653437689861552785, 1.63272357936029339660940403376, 2.34583244545989611355087103303, 2.99941437940098712138765854096, 3.64826604965760821530726335042, 3.70435541571971822099748167322, 3.88782305896589818564941135966, 4.54545014782265603366854791153, 4.85512283254279582558561488230, 5.62068913980564779903495376901, 6.20094236439109133333596148959, 6.79661597729253634860739013116, 6.97231192698558333316844758012, 7.27784894919030119363644743187
