| L(s) = 1 | + 4·5-s − 9-s − 4·11-s − 4·19-s + 11·25-s − 8·29-s + 16·31-s + 12·41-s − 4·45-s + 10·49-s − 16·55-s + 12·59-s + 16·61-s + 32·79-s + 81-s − 20·89-s − 16·95-s + 4·99-s + 24·101-s + 32·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 1.20·11-s − 0.917·19-s + 11/5·25-s − 1.48·29-s + 2.87·31-s + 1.87·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 1.56·59-s + 2.04·61-s + 3.60·79-s + 1/9·81-s − 2.11·89-s − 1.64·95-s + 0.402·99-s + 2.38·101-s + 3.06·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.106140542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.106140542\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 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
−8.528717200884972351457946038864, −8.452498587996157281697302717846, −8.036896723968698869478696504323, −7.59377727580361387070452899714, −7.06007181125636327762430039942, −6.88101584799193237081528169754, −6.28353405190586177034797264226, −6.03396248982479520978851173055, −5.74405173259605957431439558660, −5.45052804044118806616923990009, −4.85515440618035806095390842957, −4.76116300199839405429586355959, −4.11632538611497873004438514779, −3.62211787223943650633992784270, −3.04346559378314524822900113154, −2.37402884781272469210822032321, −2.36937627922177779376082395224, −2.05799618019066120815372172571, −1.02001600842478991752427651097, −0.67320986596326535404771264330,
0.67320986596326535404771264330, 1.02001600842478991752427651097, 2.05799618019066120815372172571, 2.36937627922177779376082395224, 2.37402884781272469210822032321, 3.04346559378314524822900113154, 3.62211787223943650633992784270, 4.11632538611497873004438514779, 4.76116300199839405429586355959, 4.85515440618035806095390842957, 5.45052804044118806616923990009, 5.74405173259605957431439558660, 6.03396248982479520978851173055, 6.28353405190586177034797264226, 6.88101584799193237081528169754, 7.06007181125636327762430039942, 7.59377727580361387070452899714, 8.036896723968698869478696504323, 8.452498587996157281697302717846, 8.528717200884972351457946038864
