| L(s) = 1 | + 5-s − 3·9-s + 2·11-s + 4·13-s − 17-s − 5·19-s + 9·23-s − 4·25-s − 6·29-s − 8·31-s − 11·37-s + 4·41-s − 43-s − 3·45-s + 2·47-s + 6·53-s + 2·55-s − 9·59-s − 6·61-s + 4·65-s − 7·67-s + 71-s − 4·73-s − 8·79-s + 9·81-s + 4·83-s − 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.603·11-s + 1.10·13-s − 0.242·17-s − 1.14·19-s + 1.87·23-s − 4/5·25-s − 1.11·29-s − 1.43·31-s − 1.80·37-s + 0.624·41-s − 0.152·43-s − 0.447·45-s + 0.291·47-s + 0.824·53-s + 0.269·55-s − 1.17·59-s − 0.768·61-s + 0.496·65-s − 0.855·67-s + 0.118·71-s − 0.468·73-s − 0.900·79-s + 81-s + 0.439·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−7.58174623102252501884564895132, −6.88930394730036359896690789309, −6.09670564274641919026371187386, −5.69411861547708667755966361883, −4.85729152827838644368183504197, −3.83635317650688739046775043581, −3.28672913135855046011389212463, −2.20787338074424499493905030723, −1.41071811972492894507679655738, 0,
1.41071811972492894507679655738, 2.20787338074424499493905030723, 3.28672913135855046011389212463, 3.83635317650688739046775043581, 4.85729152827838644368183504197, 5.69411861547708667755966361883, 6.09670564274641919026371187386, 6.88930394730036359896690789309, 7.58174623102252501884564895132
