| L(s) = 1 | − 4·4-s + 52·13-s + 16·16-s + 108·17-s − 264·23-s − 25·25-s − 300·29-s − 356·43-s + 110·49-s − 208·52-s + 96·53-s − 1.51e3·61-s − 64·64-s − 432·68-s + 1.76e3·79-s + 1.05e3·92-s + 100·100-s − 3.80e3·101-s + 1.26e3·103-s − 2.53e3·107-s − 4.52e3·113-s + 1.20e3·116-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.10·13-s + 1/4·16-s + 1.54·17-s − 2.39·23-s − 1/5·25-s − 1.92·29-s − 1.26·43-s + 0.320·49-s − 0.554·52-s + 0.248·53-s − 3.18·61-s − 1/8·64-s − 0.770·68-s + 2.50·79-s + 1.19·92-s + 1/10·100-s − 3.74·101-s + 1.20·103-s − 2.28·107-s − 3.76·113-s + 0.960·116-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05132074789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05132074789\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 938 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6662 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 132 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37082 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24010 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 80242 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 102670 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 73658 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 758 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 577190 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 671722 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 711470 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 630650 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 942082 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1820990 T^{2} + p^{6} T^{4} \) |
| 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
−9.719116680903302097302460875917, −9.260388973204329470984309790961, −8.867725693022668906483224611098, −8.204616342891092221736657726812, −7.909901973732622320091796728925, −7.85433128227709884352282050064, −7.24723168884636210338366401499, −6.55885603090972775165401510572, −6.15500415505209135856662590209, −5.89058041177730080419526616273, −5.22616205349839959159020440973, −5.19074119240301858177977768093, −4.19058203707643701651522025903, −3.82711494684113931242819072999, −3.69491314809903160980608264118, −2.97712311487418114780225734775, −2.28671799646428740009695308217, −1.41179447605470368376360775113, −1.37429633908248835721373436574, −0.05519925980954676636600711217,
0.05519925980954676636600711217, 1.37429633908248835721373436574, 1.41179447605470368376360775113, 2.28671799646428740009695308217, 2.97712311487418114780225734775, 3.69491314809903160980608264118, 3.82711494684113931242819072999, 4.19058203707643701651522025903, 5.19074119240301858177977768093, 5.22616205349839959159020440973, 5.89058041177730080419526616273, 6.15500415505209135856662590209, 6.55885603090972775165401510572, 7.24723168884636210338366401499, 7.85433128227709884352282050064, 7.909901973732622320091796728925, 8.204616342891092221736657726812, 8.867725693022668906483224611098, 9.260388973204329470984309790961, 9.719116680903302097302460875917
