| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 2·23-s − 24-s − 4·25-s + 27-s − 28-s + 30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.182·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−14.69986832661895, −14.28244597265977, −13.55543791362227, −13.24209123810688, −12.51024049693768, −12.09249715566418, −11.48991967983206, −11.17631029835791, −10.27284129097436, −10.02366674030014, −9.457160244195382, −9.014977606322285, −8.335438748453962, −7.886834945281114, −7.626027584307095, −6.768183899814762, −6.331890637088342, −5.868546093869051, −4.792253284958957, −4.425931919562064, −3.397904619338059, −3.316452058041775, −2.331329143642279, −1.754858027948629, −0.8914482064108276, 0,
0.8914482064108276, 1.754858027948629, 2.331329143642279, 3.316452058041775, 3.397904619338059, 4.425931919562064, 4.792253284958957, 5.868546093869051, 6.331890637088342, 6.768183899814762, 7.626027584307095, 7.886834945281114, 8.335438748453962, 9.014977606322285, 9.457160244195382, 10.02366674030014, 10.27284129097436, 11.17631029835791, 11.48991967983206, 12.09249715566418, 12.51024049693768, 13.24209123810688, 13.55543791362227, 14.28244597265977, 14.69986832661895
