| L(s) = 1 | + 4-s + 4·7-s + 6·13-s − 3·16-s + 19-s − 4·25-s + 4·28-s + 6·31-s − 4·37-s − 4·43-s + 9·49-s + 6·52-s − 12·61-s − 7·64-s − 2·67-s + 76-s + 4·79-s + 24·91-s − 18·97-s − 4·100-s + 24·103-s + 14·109-s − 12·112-s − 14·121-s + 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s + 1.66·13-s − 3/4·16-s + 0.229·19-s − 4/5·25-s + 0.755·28-s + 1.07·31-s − 0.657·37-s − 0.609·43-s + 9/7·49-s + 0.832·52-s − 1.53·61-s − 7/8·64-s − 0.244·67-s + 0.114·76-s + 0.450·79-s + 2.51·91-s − 1.82·97-s − 2/5·100-s + 2.36·103-s + 1.34·109-s − 1.13·112-s − 1.27·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050939191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050939191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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
−9.890139016986196740636193293128, −9.047437862924854413132189610335, −8.777323872284348949906879054268, −8.288976171901398913161424238801, −7.73310927919809001613255931789, −7.43209658390452300299056194774, −6.48644387760280128777685872855, −6.33862363372730772659771851904, −5.55535214636854343659925063943, −4.98159191267370344305914685116, −4.37410367617512098014642619825, −3.77603612409028846884363283351, −2.92080269569751487246811468464, −1.96839889856622208964197478652, −1.33272988835247411538861095158,
1.33272988835247411538861095158, 1.96839889856622208964197478652, 2.92080269569751487246811468464, 3.77603612409028846884363283351, 4.37410367617512098014642619825, 4.98159191267370344305914685116, 5.55535214636854343659925063943, 6.33862363372730772659771851904, 6.48644387760280128777685872855, 7.43209658390452300299056194774, 7.73310927919809001613255931789, 8.288976171901398913161424238801, 8.777323872284348949906879054268, 9.047437862924854413132189610335, 9.890139016986196740636193293128
