| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·9-s − 2·11-s + 5·16-s − 8·18-s − 4·22-s + 12·23-s + 8·25-s + 4·29-s + 6·32-s − 12·36-s − 20·37-s − 16·43-s − 6·44-s + 24·46-s + 16·50-s + 16·53-s + 8·58-s + 7·64-s + 4·67-s − 4·71-s − 16·72-s − 40·74-s + 32·79-s + 7·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s − 0.603·11-s + 5/4·16-s − 1.88·18-s − 0.852·22-s + 2.50·23-s + 8/5·25-s + 0.742·29-s + 1.06·32-s − 2·36-s − 3.28·37-s − 2.43·43-s − 0.904·44-s + 3.53·46-s + 2.26·50-s + 2.19·53-s + 1.05·58-s + 7/8·64-s + 0.488·67-s − 0.474·71-s − 1.88·72-s − 4.64·74-s + 3.60·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.731708082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.731708082\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−10.31712977353414038389342658987, −9.904899840272703022667263533530, −8.909413032360724807774144812577, −8.833869790839060640010182867948, −8.588069015342010143727576601591, −8.007950184997369309371357323402, −7.31380020293928955471865897045, −7.06495967778460744130837564757, −6.50040795053666263347425379635, −6.47409897277896400238223095768, −5.46500113815737217134328084174, −5.33161816403917439284993844242, −4.90013799359119435735702638309, −4.79434884703651495756367859786, −3.67085440498159200895188491665, −3.38279488024904310793124189001, −2.99064267300105637449759917029, −2.50720231447323381580497390108, −1.78573272761699183348331402973, −0.76692540560466746681841310049,
0.76692540560466746681841310049, 1.78573272761699183348331402973, 2.50720231447323381580497390108, 2.99064267300105637449759917029, 3.38279488024904310793124189001, 3.67085440498159200895188491665, 4.79434884703651495756367859786, 4.90013799359119435735702638309, 5.33161816403917439284993844242, 5.46500113815737217134328084174, 6.47409897277896400238223095768, 6.50040795053666263347425379635, 7.06495967778460744130837564757, 7.31380020293928955471865897045, 8.007950184997369309371357323402, 8.588069015342010143727576601591, 8.833869790839060640010182867948, 8.909413032360724807774144812577, 9.904899840272703022667263533530, 10.31712977353414038389342658987
