| L(s) = 1 | + 4-s − 3·5-s + 6·11-s + 16-s + 4·19-s − 3·20-s + 4·25-s − 12·29-s + 10·31-s + 12·41-s + 6·44-s − 13·49-s − 18·55-s − 24·59-s + 16·61-s + 64-s + 4·76-s + 16·79-s − 3·80-s + 36·89-s − 12·95-s + 4·100-s + 6·101-s + 4·109-s − 12·116-s + 5·121-s + 10·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s + 1.80·11-s + 1/4·16-s + 0.917·19-s − 0.670·20-s + 4/5·25-s − 2.22·29-s + 1.79·31-s + 1.87·41-s + 0.904·44-s − 1.85·49-s − 2.42·55-s − 3.12·59-s + 2.04·61-s + 1/8·64-s + 0.458·76-s + 1.80·79-s − 0.335·80-s + 3.81·89-s − 1.23·95-s + 2/5·100-s + 0.597·101-s + 0.383·109-s − 1.11·116-s + 5/11·121-s + 0.898·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424789353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424789353\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.512741326522107701681952133776, −9.424555904372419494656746065633, −8.904247934298067766833725269707, −8.033154449173421852738549606568, −7.82723843094092867969941089213, −7.36566276940036406451155206076, −6.71508725988629390136097161445, −6.29951651895743522476705768290, −5.74395141102766134561119794703, −4.77709333112371596512170551343, −4.34568268458938072273214663423, −3.49841666048700785356624668043, −3.38549178369413487864105189864, −2.10144548480492536578842252757, −1.01053650029556092883839412457,
1.01053650029556092883839412457, 2.10144548480492536578842252757, 3.38549178369413487864105189864, 3.49841666048700785356624668043, 4.34568268458938072273214663423, 4.77709333112371596512170551343, 5.74395141102766134561119794703, 6.29951651895743522476705768290, 6.71508725988629390136097161445, 7.36566276940036406451155206076, 7.82723843094092867969941089213, 8.033154449173421852738549606568, 8.904247934298067766833725269707, 9.424555904372419494656746065633, 9.512741326522107701681952133776
