| L(s) = 1 | − 4·2-s − 22·4-s + 140·8-s + 70·9-s − 14·13-s + 195·16-s + 35·17-s − 280·18-s + 14·19-s + 56·26-s − 2.77e3·32-s − 140·34-s − 1.54e3·36-s − 56·38-s + 592·43-s + 896·47-s + 341·49-s + 308·52-s + 630·53-s + 1.00e3·59-s + 1.04e3·64-s − 2.08e3·67-s − 770·68-s + 9.80e3·72-s − 308·76-s + 2.21e3·81-s − 2.14e3·83-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.75·4-s + 6.18·8-s + 2.59·9-s − 0.298·13-s + 3.04·16-s + 0.499·17-s − 3.66·18-s + 0.169·19-s + 0.422·26-s − 15.3·32-s − 0.706·34-s − 7.12·36-s − 0.239·38-s + 2.09·43-s + 2.78·47-s + 0.994·49-s + 0.821·52-s + 1.63·53-s + 2.22·59-s + 2.03·64-s − 3.79·67-s − 1.37·68-s + 16.0·72-s − 0.464·76-s + 3.04·81-s − 2.83·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5377157475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5377157475\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2^2$ | \( 1 - 35 T + 280 p T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{4} \) |
| 3 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 341 T^{2} + 1128 T^{4} - 341 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4293 T^{2} + 7887344 T^{4} - 4293 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 7 T - 966 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 7 T + 8358 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5028 T^{2} + 32833958 T^{4} - 5028 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 1058 p T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 32165 T^{2} + 390771768 T^{4} - 32165 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 91016 T^{2} + 5345007678 T^{4} - 91016 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 169305 T^{2} + 14486361728 T^{4} - 169305 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 148 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 315 T + 59320 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 504 T + 130438 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 91480 T^{2} + 103275473118 T^{4} + 91480 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 1041 T + 609206 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3103 p T^{2} + 267650334444 T^{4} - 3103 p^{7} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 922873 T^{2} + 419875572048 T^{4} - 922873 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1820261 T^{2} + 1309129438200 T^{4} - 1820261 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1071 T + 1424962 T^{2} + 1071 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 931 T + 1363388 T^{2} - 931 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2397332 T^{2} + 2743625522598 T^{4} - 2397332 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74281463466507980020438681541, −7.53280393488377820557947793233, −7.27131531805841697546062577595, −7.20791524297240793063666619916, −7.10542716812910151827950761916, −6.43592451714184319122213817662, −6.03733315647856846933151364470, −5.71993980255115565301646189438, −5.45206549910803858362535763924, −5.34127891848696507657038678410, −5.09232922850948785225852099744, −4.59725512627940550887162010613, −4.35360548567310295969794848037, −4.23904899194757325590043955363, −4.13974817229449330424588825595, −3.95794426081707943776524239940, −3.55511012498425988048020094737, −3.19401927856690496618639427293, −2.48305447237743809422276769437, −2.17728085808049780806942296527, −1.37019278287858476303060001564, −1.28827338895280955198509093465, −1.05088804339446745709676109229, −0.73024154617894875944534371062, −0.22749992464298863856864381722,
0.22749992464298863856864381722, 0.73024154617894875944534371062, 1.05088804339446745709676109229, 1.28827338895280955198509093465, 1.37019278287858476303060001564, 2.17728085808049780806942296527, 2.48305447237743809422276769437, 3.19401927856690496618639427293, 3.55511012498425988048020094737, 3.95794426081707943776524239940, 4.13974817229449330424588825595, 4.23904899194757325590043955363, 4.35360548567310295969794848037, 4.59725512627940550887162010613, 5.09232922850948785225852099744, 5.34127891848696507657038678410, 5.45206549910803858362535763924, 5.71993980255115565301646189438, 6.03733315647856846933151364470, 6.43592451714184319122213817662, 7.10542716812910151827950761916, 7.20791524297240793063666619916, 7.27131531805841697546062577595, 7.53280393488377820557947793233, 7.74281463466507980020438681541
