| L(s) = 1 | + 2·2-s − 3-s + 3·4-s + 2·5-s − 2·6-s + 2·7-s + 4·8-s − 9-s + 4·10-s − 8·11-s − 3·12-s + 5·13-s + 4·14-s − 2·15-s + 5·16-s + 2·17-s − 2·18-s − 19-s + 6·20-s − 2·21-s − 16·22-s − 2·23-s − 4·24-s + 3·25-s + 10·26-s + 6·28-s + 29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s − 0.816·6-s + 0.755·7-s + 1.41·8-s − 1/3·9-s + 1.26·10-s − 2.41·11-s − 0.866·12-s + 1.38·13-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 0.485·17-s − 0.471·18-s − 0.229·19-s + 1.34·20-s − 0.436·21-s − 3.41·22-s − 0.417·23-s − 0.816·24-s + 3/5·25-s + 1.96·26-s + 1.13·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.641718661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641718661\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 208 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 21 T + 214 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 176 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−13.13877933635486059572757260188, −12.66096512203286980387592419038, −12.10544810541229785350091760547, −11.66394494369606430051466871554, −10.92859627184035461118106568289, −10.68506036829011038462301877555, −10.49907698880799439263670598969, −9.784030126246858738436207990935, −8.711738385693752819143107421098, −8.466998658611338235868933325884, −7.52351833611557182118683950689, −7.34952816056617061765829089822, −6.29590725444563709729946214901, −5.79792626075858972069351425226, −5.41698935019301384835626648904, −5.17293503921575241435705278419, −4.27868487020009166436327455442, −3.40453406978190707729069353761, −2.58583688898183257087516876630, −1.77258610741404249167330044951,
1.77258610741404249167330044951, 2.58583688898183257087516876630, 3.40453406978190707729069353761, 4.27868487020009166436327455442, 5.17293503921575241435705278419, 5.41698935019301384835626648904, 5.79792626075858972069351425226, 6.29590725444563709729946214901, 7.34952816056617061765829089822, 7.52351833611557182118683950689, 8.466998658611338235868933325884, 8.711738385693752819143107421098, 9.784030126246858738436207990935, 10.49907698880799439263670598969, 10.68506036829011038462301877555, 10.92859627184035461118106568289, 11.66394494369606430051466871554, 12.10544810541229785350091760547, 12.66096512203286980387592419038, 13.13877933635486059572757260188
