| L(s) = 1 | − 17·3-s + 50·5-s − 98·7-s − 17·9-s − 167·11-s + 445·13-s − 850·15-s + 99·17-s − 1.33e3·19-s + 1.66e3·21-s − 290·23-s + 1.87e3·25-s + 1.36e3·27-s − 6.95e3·29-s − 1.04e4·31-s + 2.83e3·33-s − 4.90e3·35-s − 1.58e3·37-s − 7.56e3·39-s − 2.41e4·41-s − 2.04e4·43-s − 850·45-s + 619·47-s + 7.20e3·49-s − 1.68e3·51-s − 2.17e4·53-s − 8.35e3·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 0.894·5-s − 0.755·7-s − 0.0699·9-s − 0.416·11-s + 0.730·13-s − 0.975·15-s + 0.0830·17-s − 0.847·19-s + 0.824·21-s − 0.114·23-s + 3/5·25-s + 0.359·27-s − 1.53·29-s − 1.95·31-s + 0.453·33-s − 0.676·35-s − 0.190·37-s − 0.796·39-s − 2.23·41-s − 1.68·43-s − 0.0625·45-s + 0.0408·47-s + 3/7·49-s − 0.0906·51-s − 1.06·53-s − 0.372·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 17 T + 34 p^{2} T^{2} + 17 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 167 T + 308642 T^{2} + 167 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 445 T + 368060 T^{2} - 445 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 99 T + 2708724 T^{2} - 99 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1334 T + 4863326 T^{2} + 1334 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 290 T + 11924062 T^{2} + 290 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6959 T + 36208036 T^{2} + 6959 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10480 T + 81551678 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1588 T + 126661454 T^{2} + 1588 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 24110 T + 371658466 T^{2} + 24110 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 20406 T + 250108894 T^{2} + 20406 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 619 T + 321026782 T^{2} - 619 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 21794 T + 942703706 T^{2} + 21794 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1976 T + 1182594598 T^{2} - 1976 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 22614 T + 1309835722 T^{2} - 22614 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4068 T + 2602353254 T^{2} - 4068 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 46528 T + 52478722 p T^{2} + 46528 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 27096 T + 3934288606 T^{2} - 27096 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12481 T + 5254424046 T^{2} + 12481 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 100088 T + 6216298006 T^{2} + 100088 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 33798 T + 5723897898 T^{2} - 33798 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 138991 T + 20777859228 T^{2} - 138991 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.78757862110369482377627182238, −11.52803229812104622923910346531, −10.89804537911415483230386581160, −10.57712219275412756387755138493, −9.943017082671451480726323373585, −9.593889052339965682158172385552, −8.676370396353868551122308422538, −8.637776452675241912348686433635, −7.51964385017265384426652435588, −6.97297525221478391271678710561, −6.18819083616526366692859323519, −6.09016553309673930307534338366, −5.30326319075499741808588116082, −5.02038143259769050521257518987, −3.76797562002492160834721688329, −3.29642065593888974347401724661, −2.14039768334384122017672888240, −1.50255273385638433525691070459, 0, 0,
1.50255273385638433525691070459, 2.14039768334384122017672888240, 3.29642065593888974347401724661, 3.76797562002492160834721688329, 5.02038143259769050521257518987, 5.30326319075499741808588116082, 6.09016553309673930307534338366, 6.18819083616526366692859323519, 6.97297525221478391271678710561, 7.51964385017265384426652435588, 8.637776452675241912348686433635, 8.676370396353868551122308422538, 9.593889052339965682158172385552, 9.943017082671451480726323373585, 10.57712219275412756387755138493, 10.89804537911415483230386581160, 11.52803229812104622923910346531, 11.78757862110369482377627182238
