| L(s) = 1 | − 2-s + 3-s − 6-s + 7-s + 8-s + 3·9-s + 6·11-s + 8·13-s − 14-s − 16-s − 3·18-s − 2·19-s + 21-s − 6·22-s − 3·23-s + 24-s − 8·26-s + 8·27-s − 6·29-s − 8·31-s + 6·33-s − 4·37-s + 2·38-s + 8·39-s + 18·41-s − 42-s + 14·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 9-s + 1.80·11-s + 2.21·13-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s − 1.56·26-s + 1.53·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.657·37-s + 0.324·38-s + 1.28·39-s + 2.81·41-s − 0.154·42-s + 2.13·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.777786066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777786066\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−11.44531500776178172422114943476, −11.14633118662792361884377904949, −10.80464514412187505352138054054, −10.40390860085647571050119551459, −9.646881200135278145608348813606, −9.189683285765849672773553132719, −9.003532390468188680012755658569, −8.708747181862142582449605468436, −7.918198123693549389279408758495, −7.73052133363762198184662013333, −6.99614545356602163056626935680, −6.56068623140851519642977026679, −5.99059187620313463522781212705, −5.53934080598421419226546320249, −4.36038292048665956218834927761, −3.95193641127438853822237163575, −3.87121871979297141974644355668, −2.71238625560086399033449734165, −1.46375846141625953745599028381, −1.37271047232170476058201585725,
1.37271047232170476058201585725, 1.46375846141625953745599028381, 2.71238625560086399033449734165, 3.87121871979297141974644355668, 3.95193641127438853822237163575, 4.36038292048665956218834927761, 5.53934080598421419226546320249, 5.99059187620313463522781212705, 6.56068623140851519642977026679, 6.99614545356602163056626935680, 7.73052133363762198184662013333, 7.918198123693549389279408758495, 8.708747181862142582449605468436, 9.003532390468188680012755658569, 9.189683285765849672773553132719, 9.646881200135278145608348813606, 10.40390860085647571050119551459, 10.80464514412187505352138054054, 11.14633118662792361884377904949, 11.44531500776178172422114943476
