L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 4·10-s − 2·11-s + 6·12-s − 6·13-s − 4·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s − 6·19-s − 6·20-s + 4·21-s + 4·22-s − 2·23-s − 8·24-s + 3·25-s + 12·26-s + 4·27-s + 6·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.603·11-s + 1.73·12-s − 1.66·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 1.37·19-s − 1.34·20-s + 0.872·21-s + 0.852·22-s − 0.417·23-s − 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 118 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.86354272231428348680156130691, −7.78614953563936162887314874674, −7.39028888963835754331503056706, −7.32778435837653176903781849563, −6.73573753377005523858574591547, −6.45347128193262451852208633446, −5.92943927286570104718926828240, −5.48698972548728584931224410300, −4.75962679947219534153853845031, −4.75601841967884716540506891250, −4.25108878134129653544110239356, −3.86776990723684138904963576849, −3.09865432756061259663080502397, −2.94906274991163531189299227834, −2.45579756765848827196713573258, −2.21667261034562502990386025769, −1.46567973326919740923751247913, −1.21633840861317687266990410239, 0, 0,
1.21633840861317687266990410239, 1.46567973326919740923751247913, 2.21667261034562502990386025769, 2.45579756765848827196713573258, 2.94906274991163531189299227834, 3.09865432756061259663080502397, 3.86776990723684138904963576849, 4.25108878134129653544110239356, 4.75601841967884716540506891250, 4.75962679947219534153853845031, 5.48698972548728584931224410300, 5.92943927286570104718926828240, 6.45347128193262451852208633446, 6.73573753377005523858574591547, 7.32778435837653176903781849563, 7.39028888963835754331503056706, 7.78614953563936162887314874674, 7.86354272231428348680156130691