L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 4·8-s + 6·9-s + 4·10-s − 4·13-s + 8·16-s + 8·17-s − 12·18-s − 4·20-s + 5·25-s + 8·26-s − 4·29-s − 8·32-s − 16·34-s + 12·36-s + 26·37-s + 8·40-s + 12·41-s − 12·45-s − 10·50-s − 8·52-s + 8·58-s − 10·61-s + 8·64-s + 8·65-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 1.41·8-s + 2·9-s + 1.26·10-s − 1.10·13-s + 2·16-s + 1.94·17-s − 2.82·18-s − 0.894·20-s + 25-s + 1.56·26-s − 0.742·29-s − 1.41·32-s − 2.74·34-s + 2·36-s + 4.27·37-s + 1.26·40-s + 1.87·41-s − 1.78·45-s − 1.41·50-s − 1.10·52-s + 1.05·58-s − 1.28·61-s + 64-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8067590381\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8067590381\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
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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
−8.953236415234575795022106149282, −8.257284172969588699169807666395, −8.087281306724333358417571994698, −7.960435101760584684681433191283, −7.81162903330237251991997867934, −7.50541068453835228213439233631, −7.35449892138689520270649540817, −7.25481952305200153248340724181, −6.55673215842408077089544543449, −6.37669541986349298918186331813, −6.37636502248263064154179056782, −5.94420706766445501274002033061, −5.34750416341680038368940956355, −5.16998204508883783956200852453, −5.11092127693518215266823232705, −4.27956933075372332385803453637, −4.17525034156531332809721922749, −4.05485826838357645230918986173, −3.62800955381732527478041210188, −2.96835419636652214984410297808, −2.73711572044322894196213646765, −2.53068755208615172929797953172, −1.67014570993262806063322820643, −1.10055751900459816206783674902, −0.75224477489203201095667360963,
0.75224477489203201095667360963, 1.10055751900459816206783674902, 1.67014570993262806063322820643, 2.53068755208615172929797953172, 2.73711572044322894196213646765, 2.96835419636652214984410297808, 3.62800955381732527478041210188, 4.05485826838357645230918986173, 4.17525034156531332809721922749, 4.27956933075372332385803453637, 5.11092127693518215266823232705, 5.16998204508883783956200852453, 5.34750416341680038368940956355, 5.94420706766445501274002033061, 6.37636502248263064154179056782, 6.37669541986349298918186331813, 6.55673215842408077089544543449, 7.25481952305200153248340724181, 7.35449892138689520270649540817, 7.50541068453835228213439233631, 7.81162903330237251991997867934, 7.960435101760584684681433191283, 8.087281306724333358417571994698, 8.257284172969588699169807666395, 8.953236415234575795022106149282