L(s) = 1 | − 5·5-s + 10·25-s − 5·49-s − 5·59-s − 5·67-s − 5·103-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 5·5-s + 10·25-s − 5·49-s − 5·59-s − 5·67-s − 5·103-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{60} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01950430293\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01950430293\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 11 | \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \) |
good | 2 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 3 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 13 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 17 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 19 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 23 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 29 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 31 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 37 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 41 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \) |
| 47 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 53 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 59 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 61 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 67 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 71 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 73 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 79 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 83 | \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \) |
| 89 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
| 97 | \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.46256272595576223407420774081, −2.23674837432607368230048325067, −2.23338645253155605948969509622, −2.20348861378972512932701772302, −2.08619148406347412890827850148, −2.06703270719170011411174762041, −2.04133171884518549813673489657, −1.84112987206505368754469864201, −1.74953871968524199025171124667, −1.61731323920266975435507739687, −1.54599592651136661385259086604, −1.53408678243934952776277110015, −1.50653561665538077210631884571, −1.47188784177027240342647154405, −1.47040419388496241360827868700, −1.46247967798656412897395712023, −1.45301004557504448097286585518, −1.40430755150021376517611223190, −0.906118023088098211926350677781, −0.898481222263506974973620214188, −0.883528886354449571503620346403, −0.76271225145232390612741148376, −0.72409773050268769876074204520, −0.28609424859910201703028926055, −0.19351589102559275432015516175,
0.19351589102559275432015516175, 0.28609424859910201703028926055, 0.72409773050268769876074204520, 0.76271225145232390612741148376, 0.883528886354449571503620346403, 0.898481222263506974973620214188, 0.906118023088098211926350677781, 1.40430755150021376517611223190, 1.45301004557504448097286585518, 1.46247967798656412897395712023, 1.47040419388496241360827868700, 1.47188784177027240342647154405, 1.50653561665538077210631884571, 1.53408678243934952776277110015, 1.54599592651136661385259086604, 1.61731323920266975435507739687, 1.74953871968524199025171124667, 1.84112987206505368754469864201, 2.04133171884518549813673489657, 2.06703270719170011411174762041, 2.08619148406347412890827850148, 2.20348861378972512932701772302, 2.23338645253155605948969509622, 2.23674837432607368230048325067, 2.46256272595576223407420774081
Plot not available for L-functions of degree greater than 10.