| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s − 6·13-s + 15-s + 16-s + 2·17-s − 18-s + 20-s − 22-s − 24-s + 25-s + 6·26-s + 27-s + 10·29-s − 30-s − 32-s + 33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.66·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 1.85·29-s − 0.182·30-s − 0.176·32-s + 0.174·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.85147260729747, −13.48899917059961, −12.63780308048080, −12.37056394027074, −11.97579496351723, −11.37991919362407, −10.72387386584109, −10.16781870239002, −9.870648792312370, −9.537881699747397, −8.930115701399986, −8.386159442362767, −8.035205947142896, −7.424806393095496, −6.832069144969776, −6.647519054238419, −5.834195065991381, −5.109260306315396, −4.829777182230496, −3.997585498066373, −3.319273291270560, −2.676018172283585, −2.322865319297066, −1.579299195562949, −0.9265255308484643, 0,
0.9265255308484643, 1.579299195562949, 2.322865319297066, 2.676018172283585, 3.319273291270560, 3.997585498066373, 4.829777182230496, 5.109260306315396, 5.834195065991381, 6.647519054238419, 6.832069144969776, 7.424806393095496, 8.035205947142896, 8.386159442362767, 8.930115701399986, 9.537881699747397, 9.870648792312370, 10.16781870239002, 10.72387386584109, 11.37991919362407, 11.97579496351723, 12.37056394027074, 12.63780308048080, 13.48899917059961, 13.85147260729747
