| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·11-s − 8·13-s + 4·14-s + 5·16-s + 2·17-s − 2·19-s − 8·22-s + 16·26-s − 6·28-s − 4·29-s + 10·31-s − 6·32-s − 4·34-s + 4·38-s − 2·41-s − 4·43-s + 12·44-s − 4·47-s + 6·49-s − 24·52-s − 8·53-s + 8·56-s + 8·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1.20·11-s − 2.21·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 0.458·19-s − 1.70·22-s + 3.13·26-s − 1.13·28-s − 0.742·29-s + 1.79·31-s − 1.06·32-s − 0.685·34-s + 0.648·38-s − 0.312·41-s − 0.609·43-s + 1.80·44-s − 0.583·47-s + 6/7·49-s − 3.32·52-s − 1.09·53-s + 1.06·56-s + 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 270 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−7.53065022155542769193869380124, −7.26136194025217492800906837101, −7.00081772948116793776792153086, −6.81855703903069503138138871376, −6.34078914137492780766939385891, −6.04265504084003593442577180819, −5.76859424604496967109365279592, −5.19136465650262397296598270495, −4.77562456258633642038820441409, −4.55319342737240920408136173229, −3.92329947105747106468742342945, −3.61634872218263538465220191845, −3.03297692689634641314716686881, −2.82231134717789269539524552329, −2.26749616285473194363629563974, −2.02033494488443966428932173692, −1.30240109854654402059735310272, −0.982780567447876927227737313937, 0, 0,
0.982780567447876927227737313937, 1.30240109854654402059735310272, 2.02033494488443966428932173692, 2.26749616285473194363629563974, 2.82231134717789269539524552329, 3.03297692689634641314716686881, 3.61634872218263538465220191845, 3.92329947105747106468742342945, 4.55319342737240920408136173229, 4.77562456258633642038820441409, 5.19136465650262397296598270495, 5.76859424604496967109365279592, 6.04265504084003593442577180819, 6.34078914137492780766939385891, 6.81855703903069503138138871376, 7.00081772948116793776792153086, 7.26136194025217492800906837101, 7.53065022155542769193869380124
