| L(s) = 1 | − 4-s + 4·5-s + 7-s − 4·9-s − 2·13-s − 3·16-s − 4·20-s + 2·23-s + 2·25-s − 28-s + 4·29-s + 4·35-s + 4·36-s − 16·45-s − 4·49-s + 2·52-s + 12·53-s + 2·59-s − 4·63-s + 7·64-s − 8·65-s − 14·67-s − 12·80-s + 7·81-s − 12·83-s − 2·91-s − 2·92-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s + 0.377·7-s − 4/3·9-s − 0.554·13-s − 3/4·16-s − 0.894·20-s + 0.417·23-s + 2/5·25-s − 0.188·28-s + 0.742·29-s + 0.676·35-s + 2/3·36-s − 2.38·45-s − 4/7·49-s + 0.277·52-s + 1.64·53-s + 0.260·59-s − 0.503·63-s + 7/8·64-s − 0.992·65-s − 1.71·67-s − 1.34·80-s + 7/9·81-s − 1.31·83-s − 0.209·91-s − 0.208·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9419312153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9419312153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
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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
−11.83339669180257609565944302858, −11.68535507013654368859848359878, −10.81692738288498619970753615675, −10.23436050746063358406234314273, −9.712870704337976178143515135088, −9.126714636661016225410267328789, −8.733856836143840569093231414060, −8.043549391330799288857236799773, −7.10157501554749244505029850065, −6.30408672236389073866960478832, −5.68781909503681754640411478581, −5.22140645748655013566237515271, −4.35932945137902958767924049257, −2.90866313208207034385166985163, −2.05561271035874881637416218836,
2.05561271035874881637416218836, 2.90866313208207034385166985163, 4.35932945137902958767924049257, 5.22140645748655013566237515271, 5.68781909503681754640411478581, 6.30408672236389073866960478832, 7.10157501554749244505029850065, 8.043549391330799288857236799773, 8.733856836143840569093231414060, 9.126714636661016225410267328789, 9.712870704337976178143515135088, 10.23436050746063358406234314273, 10.81692738288498619970753615675, 11.68535507013654368859848359878, 11.83339669180257609565944302858
