| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s − 2·9-s − 12-s + 13-s + 3·14-s + 16-s − 2·18-s − 3·21-s − 4·23-s − 24-s + 26-s + 5·27-s + 3·28-s + 4·29-s − 5·31-s + 32-s − 2·36-s − 8·37-s − 39-s − 3·42-s + 4·43-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.471·18-s − 0.654·21-s − 0.834·23-s − 0.204·24-s + 0.196·26-s + 0.962·27-s + 0.566·28-s + 0.742·29-s − 0.898·31-s + 0.176·32-s − 1/3·36-s − 1.31·37-s − 0.160·39-s − 0.462·42-s + 0.609·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.35085316340932, −15.83024721055302, −15.20981698713715, −14.50980185709336, −14.24041030977443, −13.74541990642921, −13.00775858251171, −12.29807551197701, −11.91595742809191, −11.36885911559972, −10.87269803913090, −10.46432820954024, −9.593721013591670, −8.740705120951904, −8.242305029579406, −7.653617152982792, −6.890818705488640, −6.164718104622557, −5.710042344030263, −4.970589052494035, −4.622983574641899, −3.701712785286155, −2.986069227384926, −2.027435020636368, −1.331966283158759, 0,
1.331966283158759, 2.027435020636368, 2.986069227384926, 3.701712785286155, 4.622983574641899, 4.970589052494035, 5.710042344030263, 6.164718104622557, 6.890818705488640, 7.653617152982792, 8.242305029579406, 8.740705120951904, 9.593721013591670, 10.46432820954024, 10.87269803913090, 11.36885911559972, 11.91595742809191, 12.29807551197701, 13.00775858251171, 13.74541990642921, 14.24041030977443, 14.50980185709336, 15.20981698713715, 15.83024721055302, 16.35085316340932
