| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 4·13-s + 16-s − 6·17-s − 18-s + 19-s + 24-s − 4·26-s − 27-s + 2·29-s − 32-s + 6·34-s + 36-s − 4·37-s − 38-s − 4·39-s + 12·41-s − 6·43-s − 48-s − 7·49-s + 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.371·29-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-s − 0.162·38-s − 0.640·39-s + 1.87·41-s − 0.914·43-s − 0.144·48-s − 49-s + 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5052276637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5052276637\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.44816608400097, −12.12591826213692, −11.31476919545893, −11.12229166691374, −10.96388507722731, −10.36173876577383, −9.757877707484300, −9.408056239150423, −8.964220434387266, −8.381733350044072, −8.116665550047163, −7.521896370745134, −6.872055268334341, −6.608371220788155, −6.139289279026247, −5.700774836770593, −5.060949462180263, −4.494222457576395, −4.085650712816632, −3.362582202252803, −2.870465375737696, −2.162704702361743, −1.539675733925766, −1.118482572963318, −0.2312152367974436,
0.2312152367974436, 1.118482572963318, 1.539675733925766, 2.162704702361743, 2.870465375737696, 3.362582202252803, 4.085650712816632, 4.494222457576395, 5.060949462180263, 5.700774836770593, 6.139289279026247, 6.608371220788155, 6.872055268334341, 7.521896370745134, 8.116665550047163, 8.381733350044072, 8.964220434387266, 9.408056239150423, 9.757877707484300, 10.36173876577383, 10.96388507722731, 11.12229166691374, 11.31476919545893, 12.12591826213692, 12.44816608400097
