| L(s) = 1 | + 3-s + 3·7-s + 9-s + 5·11-s + 6·13-s + 5·19-s + 3·21-s − 4·23-s + 27-s + 6·29-s + 5·31-s + 5·33-s − 7·37-s + 6·39-s − 10·41-s − 9·43-s − 7·47-s + 2·49-s + 9·53-s + 5·57-s − 10·61-s + 3·63-s + 13·67-s − 4·69-s − 10·71-s + 16·73-s + 15·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.66·13-s + 1.14·19-s + 0.654·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s + 0.898·31-s + 0.870·33-s − 1.15·37-s + 0.960·39-s − 1.56·41-s − 1.37·43-s − 1.02·47-s + 2/7·49-s + 1.23·53-s + 0.662·57-s − 1.28·61-s + 0.377·63-s + 1.58·67-s − 0.481·69-s − 1.18·71-s + 1.87·73-s + 1.70·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.937317046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.937317046\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.40440854333278, −11.89885658593839, −11.75259057073028, −11.33185736237284, −10.78121404783733, −10.18529734559928, −9.890186398883726, −9.292007886046054, −8.728350855040296, −8.357809296605620, −8.272132616955233, −7.589329503186531, −6.933660518630924, −6.525692325252574, −6.210035691268382, −5.446067811911959, −4.952290259495073, −4.495562209157656, −3.875370208812947, −3.434922770240963, −3.157842810585144, −2.122808879895665, −1.623596142205656, −1.301888419805959, −0.6961126632323009,
0.6961126632323009, 1.301888419805959, 1.623596142205656, 2.122808879895665, 3.157842810585144, 3.434922770240963, 3.875370208812947, 4.495562209157656, 4.952290259495073, 5.446067811911959, 6.210035691268382, 6.525692325252574, 6.933660518630924, 7.589329503186531, 8.272132616955233, 8.357809296605620, 8.728350855040296, 9.292007886046054, 9.890186398883726, 10.18529734559928, 10.78121404783733, 11.33185736237284, 11.75259057073028, 11.89885658593839, 12.40440854333278
