| L(s) = 1 | − 3-s − 5-s − 5·7-s − 2·11-s + 2·13-s + 15-s + 4·17-s − 19-s + 5·21-s + 4·23-s + 27-s − 5·31-s + 2·33-s + 5·35-s + 5·37-s − 2·39-s + 4·41-s + 18·43-s − 2·47-s + 18·49-s − 4·51-s − 12·53-s + 2·55-s + 57-s − 8·59-s + 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s − 0.603·11-s + 0.554·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s + 1.09·21-s + 0.834·23-s + 0.192·27-s − 0.898·31-s + 0.348·33-s + 0.845·35-s + 0.821·37-s − 0.320·39-s + 0.624·41-s + 2.74·43-s − 0.291·47-s + 18/7·49-s − 0.560·51-s − 1.64·53-s + 0.269·55-s + 0.132·57-s − 1.04·59-s + 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057873300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057873300\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.372960511496372624888970697284, −9.203159774340059391327469389048, −9.068474821579911004764460608243, −8.288658907916360294317972083189, −7.79137326155969896130883151046, −7.64386792658997494728107268695, −7.16411723315561478427875274431, −6.62954887090789796740107830443, −6.22140240530667056451621708955, −6.12311683800308626604697245279, −5.35945929500864102916580883513, −5.34989908838587901097000897222, −4.54253484675341421123515623297, −4.00033967287986069340874418936, −3.63010181154341995121709656309, −3.16106563705712369170331132160, −2.73511992278929685097364155776, −2.13999297252676775047634171269, −0.986972068742934260807236016782, −0.51004141795929321380113246737,
0.51004141795929321380113246737, 0.986972068742934260807236016782, 2.13999297252676775047634171269, 2.73511992278929685097364155776, 3.16106563705712369170331132160, 3.63010181154341995121709656309, 4.00033967287986069340874418936, 4.54253484675341421123515623297, 5.34989908838587901097000897222, 5.35945929500864102916580883513, 6.12311683800308626604697245279, 6.22140240530667056451621708955, 6.62954887090789796740107830443, 7.16411723315561478427875274431, 7.64386792658997494728107268695, 7.79137326155969896130883151046, 8.288658907916360294317972083189, 9.068474821579911004764460608243, 9.203159774340059391327469389048, 9.372960511496372624888970697284
