| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 3·7-s + 8-s − 2·9-s + 10-s − 11-s + 12-s + 6·13-s − 3·14-s + 15-s + 16-s + 7·17-s − 2·18-s − 5·19-s + 20-s − 3·21-s − 22-s − 6·23-s + 24-s + 25-s + 6·26-s − 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.42429618672285, −12.94850379751033, −12.62611810093001, −12.14638634244338, −11.46469994913858, −11.12399543365304, −10.38832276982777, −10.25395151527553, −9.534289765496091, −9.108081115193346, −8.646752971475770, −8.019962883836026, −7.718657803158968, −7.048841687000316, −6.251817266144175, −6.016661551747727, −5.802259421205452, −5.178429541352868, −4.234795966776496, −3.819866699028793, −3.305237271162647, −3.043097962874911, −2.225028322917676, −1.793539934676028, −0.9198248606864496, 0,
0.9198248606864496, 1.793539934676028, 2.225028322917676, 3.043097962874911, 3.305237271162647, 3.819866699028793, 4.234795966776496, 5.178429541352868, 5.802259421205452, 6.016661551747727, 6.251817266144175, 7.048841687000316, 7.718657803158968, 8.019962883836026, 8.646752971475770, 9.108081115193346, 9.534289765496091, 10.25395151527553, 10.38832276982777, 11.12399543365304, 11.46469994913858, 12.14638634244338, 12.62611810093001, 12.94850379751033, 13.42429618672285
