| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s + 4·13-s + 16-s + 17-s − 18-s − 2·19-s + 6·22-s + 2·23-s − 24-s − 4·26-s + 27-s − 4·29-s − 32-s − 6·33-s − 34-s + 36-s + 2·37-s + 2·38-s + 4·39-s + 6·41-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 − 1.80·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s + 1.27·22-s + 0.417·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.742·29-s − 0.176·32-s − 1.04·33-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.324·38-s + 0.640·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169408811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169408811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−13.33384939418019, −13.05729239869194, −12.76637401382054, −12.10633730710162, −11.24116685382429, −11.12091795087492, −10.60102296950528, −9.961842098413215, −9.797039716518364, −8.978885104659343, −8.567022120799936, −8.256339259738534, −7.591906279274872, −7.402327375896566, −6.673735159614730, −5.979885041688968, −5.649591994683203, −4.930580320270919, −4.319658627201593, −3.623916442379755, −2.943957737247043, −2.667955461704338, −1.830466295718717, −1.314788359991883, −0.3518535139110874,
0.3518535139110874, 1.314788359991883, 1.830466295718717, 2.667955461704338, 2.943957737247043, 3.623916442379755, 4.319658627201593, 4.930580320270919, 5.649591994683203, 5.979885041688968, 6.673735159614730, 7.402327375896566, 7.591906279274872, 8.256339259738534, 8.567022120799936, 8.978885104659343, 9.797039716518364, 9.961842098413215, 10.60102296950528, 11.12091795087492, 11.24116685382429, 12.10633730710162, 12.76637401382054, 13.05729239869194, 13.33384939418019
