| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 4·11-s + 2·13-s − 4·14-s + 5·16-s − 2·17-s + 6·19-s + 8·22-s + 4·23-s − 4·26-s + 6·28-s − 10·29-s − 6·32-s + 4·34-s − 4·37-s − 12·38-s − 8·41-s − 12·44-s − 8·46-s + 2·47-s + 3·49-s + 6·52-s − 6·53-s − 8·56-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.37·19-s + 1.70·22-s + 0.834·23-s − 0.784·26-s + 1.13·28-s − 1.85·29-s − 1.06·32-s + 0.685·34-s − 0.657·37-s − 1.94·38-s − 1.24·41-s − 1.80·44-s − 1.17·46-s + 0.291·47-s + 3/7·49-s + 0.832·52-s − 0.824·53-s − 1.06·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 73 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 105 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 161 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 93 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−7.49506321546237785582283663427, −7.40192135823526484139726646766, −6.91194857756771095215938476320, −6.86462405734080933227230339599, −6.05704680829244059534920035967, −5.99521021703325236608977722028, −5.51332439630514213567480174561, −5.12679276688210602690092288192, −4.95450945810132164307489500855, −4.49566717085629860074630671956, −3.84286672613746588369990259052, −3.50251977812720071468069635006, −3.06091606700654577590141770348, −2.82332828624089217476612858898, −2.07422030889149788830524973187, −1.98722147085596455364368312127, −1.20633958256236111661829908986, −1.18477920583352514228386994337, 0, 0,
1.18477920583352514228386994337, 1.20633958256236111661829908986, 1.98722147085596455364368312127, 2.07422030889149788830524973187, 2.82332828624089217476612858898, 3.06091606700654577590141770348, 3.50251977812720071468069635006, 3.84286672613746588369990259052, 4.49566717085629860074630671956, 4.95450945810132164307489500855, 5.12679276688210602690092288192, 5.51332439630514213567480174561, 5.99521021703325236608977722028, 6.05704680829244059534920035967, 6.86462405734080933227230339599, 6.91194857756771095215938476320, 7.40192135823526484139726646766, 7.49506321546237785582283663427
