| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 17-s + 4·19-s + 25-s + 27-s − 6·29-s − 4·31-s − 4·33-s + 2·37-s − 2·39-s − 2·41-s − 4·43-s − 45-s − 12·47-s − 7·49-s − 51-s − 6·53-s + 4·55-s + 4·57-s + 4·59-s − 10·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.242·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s − 49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.681991574443395508368275921269, −7.80678815537297375192541877351, −7.51845127809073805375813598057, −6.51502238579970807104013980985, −5.34521820359272795877566829628, −4.74793032395911930853838762612, −3.58318145595154433556424126041, −2.86978163344346392409675414869, −1.76817812215837218249583925630, 0,
1.76817812215837218249583925630, 2.86978163344346392409675414869, 3.58318145595154433556424126041, 4.74793032395911930853838762612, 5.34521820359272795877566829628, 6.51502238579970807104013980985, 7.51845127809073805375813598057, 7.80678815537297375192541877351, 8.681991574443395508368275921269
