| L(s) = 1 | − 2·3-s − 4-s − 2·7-s + 3·9-s − 2·11-s + 2·12-s + 16-s + 4·21-s − 25-s − 4·27-s + 2·28-s + 4·33-s − 3·36-s + 12·37-s − 4·41-s + 2·44-s + 24·47-s − 2·48-s − 11·49-s + 22·53-s − 6·63-s − 64-s + 24·67-s + 12·71-s − 2·73-s + 2·75-s + 4·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.755·7-s + 9-s − 0.603·11-s + 0.577·12-s + 1/4·16-s + 0.872·21-s − 1/5·25-s − 0.769·27-s + 0.377·28-s + 0.696·33-s − 1/2·36-s + 1.97·37-s − 0.624·41-s + 0.301·44-s + 3.50·47-s − 0.288·48-s − 1.57·49-s + 3.02·53-s − 0.755·63-s − 1/8·64-s + 2.93·67-s + 1.42·71-s − 0.234·73-s + 0.230·75-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8573701856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8573701856\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−10.02592590252124287993361174379, −9.757674592322395651360084876713, −9.262152701564750930883455918575, −9.023074740307522681850076441493, −8.303894586725900257113948758249, −8.020486131892934550098755552225, −7.52301015595131319283184415607, −6.99416720499669562100549753595, −6.72400727434630523911025153399, −6.19429555039558421062594892004, −5.66328319385818833104268411075, −5.51968528325553463368885747218, −4.96389623615838547188196016628, −4.46116264552764782255529051814, −3.78866922353619549736702948144, −3.73114384413749356116416006896, −2.56023766661685018531833096121, −2.38677319813039334606711147299, −1.11520569429418792601494087728, −0.52536425682019317544191150121,
0.52536425682019317544191150121, 1.11520569429418792601494087728, 2.38677319813039334606711147299, 2.56023766661685018531833096121, 3.73114384413749356116416006896, 3.78866922353619549736702948144, 4.46116264552764782255529051814, 4.96389623615838547188196016628, 5.51968528325553463368885747218, 5.66328319385818833104268411075, 6.19429555039558421062594892004, 6.72400727434630523911025153399, 6.99416720499669562100549753595, 7.52301015595131319283184415607, 8.020486131892934550098755552225, 8.303894586725900257113948758249, 9.023074740307522681850076441493, 9.262152701564750930883455918575, 9.757674592322395651360084876713, 10.02592590252124287993361174379
