| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s − 2·9-s + 10-s − 12-s − 5·13-s + 15-s + 16-s + 17-s + 2·18-s + 19-s − 20-s + 6·23-s + 24-s + 25-s + 5·26-s + 5·27-s − 9·29-s − 30-s + 31-s − 32-s − 34-s − 2·36-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 − 2/3·9-s + 0.316·10-s − 0.288·12-s − 1.38·13-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.962·27-s − 1.67·29-s − 0.182·30-s + 0.179·31-s − 0.176·32-s − 0.171·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.48393276005101874941263655864, −6.96411315091461743717426172373, −6.13973693669212759093846671075, −5.35103776353047269473020794511, −4.90472192242998809246951568760, −3.79932271291007241566819443830, −2.93175713381849537629202983492, −2.19513665356035202330167405959, −0.909724936546447536782092778555, 0,
0.909724936546447536782092778555, 2.19513665356035202330167405959, 2.93175713381849537629202983492, 3.79932271291007241566819443830, 4.90472192242998809246951568760, 5.35103776353047269473020794511, 6.13973693669212759093846671075, 6.96411315091461743717426172373, 7.48393276005101874941263655864
