| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s − 5·13-s − 6·17-s + 6·19-s − 6·21-s + 23-s − 5·25-s + 9·27-s + 9·29-s + 3·31-s − 8·37-s − 15·39-s + 3·41-s − 8·43-s + 7·47-s − 3·49-s − 18·51-s − 2·53-s + 18·57-s + 4·59-s − 10·61-s − 12·63-s + 8·67-s + 3·69-s + 7·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s − 1.38·13-s − 1.45·17-s + 1.37·19-s − 1.30·21-s + 0.208·23-s − 25-s + 1.73·27-s + 1.67·29-s + 0.538·31-s − 1.31·37-s − 2.40·39-s + 0.468·41-s − 1.21·43-s + 1.02·47-s − 3/7·49-s − 2.52·51-s − 0.274·53-s + 2.38·57-s + 0.520·59-s − 1.28·61-s − 1.51·63-s + 0.977·67-s + 0.361·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.753106746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753106746\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.87278196363648315072839026373, −11.87035883342760440963614144523, −10.16281162238176530493877849897, −9.541607038853315592369443150954, −8.669848375640950008472730376997, −7.60364598197277336422475261193, −6.71449048335407632069309819908, −4.73629920554986008413543422689, −3.35238603100137584697524706621, −2.33708962511385070691012062306,
2.33708962511385070691012062306, 3.35238603100137584697524706621, 4.73629920554986008413543422689, 6.71449048335407632069309819908, 7.60364598197277336422475261193, 8.669848375640950008472730376997, 9.541607038853315592369443150954, 10.16281162238176530493877849897, 11.87035883342760440963614144523, 12.87278196363648315072839026373
