| L(s) = 1 | − 2·3-s − 4-s − 6·7-s + 3·9-s + 2·12-s + 16-s + 2·17-s − 10·19-s + 12·21-s − 8·23-s − 4·27-s + 6·28-s − 3·36-s + 14·37-s − 2·48-s + 13·49-s − 4·51-s + 20·57-s − 18·63-s − 64-s − 2·68-s + 16·69-s + 32·73-s + 10·76-s + 5·81-s − 12·84-s + 20·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 2.26·7-s + 9-s + 0.577·12-s + 1/4·16-s + 0.485·17-s − 2.29·19-s + 2.61·21-s − 1.66·23-s − 0.769·27-s + 1.13·28-s − 1/2·36-s + 2.30·37-s − 0.288·48-s + 13/7·49-s − 0.560·51-s + 2.64·57-s − 2.26·63-s − 1/8·64-s − 0.242·68-s + 1.92·69-s + 3.74·73-s + 1.14·76-s + 5/9·81-s − 1.30·84-s + 2.11·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4812636780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4812636780\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.414927164297947980608749563504, −8.773551387398185707761943250273, −8.237248423204515699602225244057, −7.989451479938124513643972394888, −7.58079550310007058586264072302, −6.95789300510819803774295775428, −6.54395010989131009805829933896, −6.36788197621047288192458756420, −5.99347043961646965132688751069, −5.93392284847000430751674107431, −5.19075129015331250218788574717, −4.77886472546419569541379311870, −4.20778172632048047723647360780, −3.97970017231065183016795723018, −3.55253931896825744238998611583, −3.02990846537475822277519587076, −2.31052781520492983081555739677, −1.95982928353895871200462675414, −0.78223713197365785793932615278, −0.35641969466039030127234739889,
0.35641969466039030127234739889, 0.78223713197365785793932615278, 1.95982928353895871200462675414, 2.31052781520492983081555739677, 3.02990846537475822277519587076, 3.55253931896825744238998611583, 3.97970017231065183016795723018, 4.20778172632048047723647360780, 4.77886472546419569541379311870, 5.19075129015331250218788574717, 5.93392284847000430751674107431, 5.99347043961646965132688751069, 6.36788197621047288192458756420, 6.54395010989131009805829933896, 6.95789300510819803774295775428, 7.58079550310007058586264072302, 7.989451479938124513643972394888, 8.237248423204515699602225244057, 8.773551387398185707761943250273, 9.414927164297947980608749563504
