| L(s) = 1 | + 4·2-s + 10·4-s − 3·5-s + 2·7-s + 20·8-s − 12·10-s + 4·13-s + 8·14-s + 35·16-s − 30·20-s + 5·25-s + 16·26-s + 20·28-s + 12·29-s + 56·32-s − 6·35-s + 2·37-s − 60·40-s − 30·47-s − 9·49-s + 20·50-s + 40·52-s + 40·56-s + 48·58-s − 4·61-s + 84·64-s − 12·65-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 1.34·5-s + 0.755·7-s + 7.07·8-s − 3.79·10-s + 1.10·13-s + 2.13·14-s + 35/4·16-s − 6.70·20-s + 25-s + 3.13·26-s + 3.77·28-s + 2.22·29-s + 9.89·32-s − 1.01·35-s + 0.328·37-s − 9.48·40-s − 4.37·47-s − 9/7·49-s + 2.82·50-s + 5.54·52-s + 5.34·56-s + 6.30·58-s − 0.512·61-s + 21/2·64-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.27266736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.27266736\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 528 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1926 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3648 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 13950 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 109 T^{2} + 7896 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 34254 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.02300038509192239584902304461, −6.38429223007674237283851332899, −6.32795012203632366649177504547, −6.29270974862386928888331349349, −6.27680924077573377935969328406, −5.96883977065153082202588299649, −5.50072022793896028903818546724, −5.25114421546138453709725781650, −5.11722686260366455222119066140, −4.77010282037161730301247026070, −4.63066114943527800258670149987, −4.57629479210009432335293362898, −4.35143874021737211281517772561, −4.06858742627771019798536967582, −3.78611823811922250600216231438, −3.35781664290822865036456478316, −3.26446806177464618334641124168, −3.23265346145596407412159520103, −2.82720458933667887836289108113, −2.65266822577909214474774025322, −2.15753349596191078987437937529, −1.57031551118256245622315699212, −1.55533334390562633653272221645, −1.32378813071448046672401272577, −0.46922539166368978251256574118,
0.46922539166368978251256574118, 1.32378813071448046672401272577, 1.55533334390562633653272221645, 1.57031551118256245622315699212, 2.15753349596191078987437937529, 2.65266822577909214474774025322, 2.82720458933667887836289108113, 3.23265346145596407412159520103, 3.26446806177464618334641124168, 3.35781664290822865036456478316, 3.78611823811922250600216231438, 4.06858742627771019798536967582, 4.35143874021737211281517772561, 4.57629479210009432335293362898, 4.63066114943527800258670149987, 4.77010282037161730301247026070, 5.11722686260366455222119066140, 5.25114421546138453709725781650, 5.50072022793896028903818546724, 5.96883977065153082202588299649, 6.27680924077573377935969328406, 6.29270974862386928888331349349, 6.32795012203632366649177504547, 6.38429223007674237283851332899, 7.02300038509192239584902304461
