| L(s) = 1 | − 24·5-s − 8·7-s + 36·11-s − 26·13-s + 12·17-s − 120·19-s + 72·23-s + 222·25-s − 84·29-s − 368·31-s + 192·35-s + 156·37-s + 216·41-s − 104·43-s − 660·47-s − 278·49-s + 1.23e3·53-s − 864·55-s − 468·59-s − 652·61-s + 624·65-s + 256·67-s − 756·71-s − 1.12e3·73-s − 288·77-s + 320·79-s + 660·83-s + ⋯ |
| L(s) = 1 | − 2.14·5-s − 0.431·7-s + 0.986·11-s − 0.554·13-s + 0.171·17-s − 1.44·19-s + 0.652·23-s + 1.77·25-s − 0.537·29-s − 2.13·31-s + 0.927·35-s + 0.693·37-s + 0.822·41-s − 0.368·43-s − 2.04·47-s − 0.810·49-s + 3.20·53-s − 2.11·55-s − 1.03·59-s − 1.36·61-s + 1.19·65-s + 0.466·67-s − 1.26·71-s − 1.80·73-s − 0.426·77-s + 0.455·79-s + 0.872·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2473881255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2473881255\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 24 T + 354 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 342 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 T + 426 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T - 3098 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 120 T + 14078 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 72 T + 15390 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 47982 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 368 T + 75798 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 156 T + 71390 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 216 T + 68506 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 104 T + 91158 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 660 T + 270306 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 618 T + p^{3} T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 468 T + 330954 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 652 T + 277998 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 256 T + 315150 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 756 T + 856146 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1124 T + 977238 T^{2} + 1124 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 320 T + 1005918 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 660 T + 1181914 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2112 T + 2524714 T^{2} - 2112 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 116 T + 1412550 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−8.942155868486544380809351552160, −8.761786900751080240072464519395, −8.056403623480995897295558024394, −8.034909988727955804996373449157, −7.33758244260778956653882935227, −7.32272971465180994437813323312, −6.81134613175450823800602574635, −6.41375570647146579173167447691, −5.88973269720466161691608412312, −5.50334943349871481484677445482, −4.67728426029417467761794525222, −4.58896235578791201598832811721, −3.97143395146052740524719131118, −3.77230498081580308723329242628, −3.33095284387081781428300304623, −2.87289498792147773975042770647, −2.06898921329447636407730059117, −1.59680315877795490941926055146, −0.72782827553517665138437509804, −0.14842948319467762912087656992,
0.14842948319467762912087656992, 0.72782827553517665138437509804, 1.59680315877795490941926055146, 2.06898921329447636407730059117, 2.87289498792147773975042770647, 3.33095284387081781428300304623, 3.77230498081580308723329242628, 3.97143395146052740524719131118, 4.58896235578791201598832811721, 4.67728426029417467761794525222, 5.50334943349871481484677445482, 5.88973269720466161691608412312, 6.41375570647146579173167447691, 6.81134613175450823800602574635, 7.32272971465180994437813323312, 7.33758244260778956653882935227, 8.034909988727955804996373449157, 8.056403623480995897295558024394, 8.761786900751080240072464519395, 8.942155868486544380809351552160
