| L(s) = 1 | − 2·3-s − 2·4-s + 5-s − 3·7-s + 4·9-s − 11-s + 4·12-s + 13-s − 2·15-s + 4·17-s − 2·20-s + 6·21-s + 6·23-s − 7·25-s − 5·27-s + 6·28-s + 2·29-s − 8·31-s + 2·33-s − 3·35-s − 8·36-s + 2·37-s − 2·39-s − 13·41-s + 2·43-s + 2·44-s + 4·45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.447·5-s − 1.13·7-s + 4/3·9-s − 0.301·11-s + 1.15·12-s + 0.277·13-s − 0.516·15-s + 0.970·17-s − 0.447·20-s + 1.30·21-s + 1.25·23-s − 7/5·25-s − 0.962·27-s + 1.13·28-s + 0.371·29-s − 1.43·31-s + 0.348·33-s − 0.507·35-s − 4/3·36-s + 0.328·37-s − 0.320·39-s − 2.03·41-s + 0.304·43-s + 0.301·44-s + 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2941146301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2941146301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 199 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 79 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 66 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 118 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 131 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 208 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.6839362141, −18.7819932193, −18.6417374867, −18.1410979991, −17.3287477302, −16.9890850725, −16.3825663743, −15.7207208258, −15.1843363171, −14.0812385401, −13.5182728901, −12.9841900333, −12.4884333510, −11.7265263303, −10.7761724556, −10.1221213866, −9.54795497494, −8.89671821587, −7.58339402007, −6.66599571585, −5.76582337619, −5.02002734030, −3.72505107797,
3.72505107797, 5.02002734030, 5.76582337619, 6.66599571585, 7.58339402007, 8.89671821587, 9.54795497494, 10.1221213866, 10.7761724556, 11.7265263303, 12.4884333510, 12.9841900333, 13.5182728901, 14.0812385401, 15.1843363171, 15.7207208258, 16.3825663743, 16.9890850725, 17.3287477302, 18.1410979991, 18.6417374867, 18.7819932193, 19.6839362141
