| L(s) = 1 | − 2·2-s − 5·4-s + 14·7-s + 20·8-s − 2·9-s + 4·11-s − 28·14-s + 5·16-s + 4·18-s − 8·22-s + 52·23-s − 5·25-s − 70·28-s − 44·29-s − 118·32-s + 10·36-s + 28·37-s − 68·43-s − 20·44-s − 104·46-s + 147·49-s + 10·50-s − 68·53-s + 280·56-s + 88·58-s − 28·63-s + 111·64-s + ⋯ |
| L(s) = 1 | − 2-s − 5/4·4-s + 2·7-s + 5/2·8-s − 2/9·9-s + 4/11·11-s − 2·14-s + 5/16·16-s + 2/9·18-s − 0.363·22-s + 2.26·23-s − 1/5·25-s − 5/2·28-s − 1.51·29-s − 3.68·32-s + 5/18·36-s + 0.756·37-s − 1.58·43-s − 0.454·44-s − 2.26·46-s + 3·49-s + 1/5·50-s − 1.28·53-s + 5·56-s + 1.51·58-s − 4/9·63-s + 1.73·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5692845435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5692845435\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )( 1 + 4 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 542 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 958 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3698 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5342 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1378 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7778 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12158 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15122 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18098 T^{2} + p^{4} T^{4} \) |
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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
−16.90062779357632364130977895505, −16.54163838919304782470716598261, −15.18054868318445563270778672281, −14.78419501369660308067686865692, −14.34737609418793252514283430911, −13.66427625720846890808078326732, −13.20739135960693087104765265339, −12.47399735256140568264933170236, −11.34461490680469755766888114602, −11.08651718063902348434576701937, −10.36326966463145767838290231595, −9.297702931867797212328224996782, −9.164014712372264624353312867507, −8.192361067925638487782006921128, −7.995953454681235411202412604988, −7.08270241161797109038839913690, −5.28862869321859185071245216462, −4.94681256561585551199079696374, −3.96071990847575989867755262542, −1.37313495336699109588713138389,
1.37313495336699109588713138389, 3.96071990847575989867755262542, 4.94681256561585551199079696374, 5.28862869321859185071245216462, 7.08270241161797109038839913690, 7.995953454681235411202412604988, 8.192361067925638487782006921128, 9.164014712372264624353312867507, 9.297702931867797212328224996782, 10.36326966463145767838290231595, 11.08651718063902348434576701937, 11.34461490680469755766888114602, 12.47399735256140568264933170236, 13.20739135960693087104765265339, 13.66427625720846890808078326732, 14.34737609418793252514283430911, 14.78419501369660308067686865692, 15.18054868318445563270778672281, 16.54163838919304782470716598261, 16.90062779357632364130977895505
