| L(s) = 1 | + 3-s − 2·9-s − 11-s − 5·13-s + 17-s + 6·19-s + 4·23-s − 5·27-s + 3·29-s − 2·31-s − 33-s − 8·37-s − 5·39-s + 10·41-s + 2·43-s − 7·47-s + 51-s + 2·53-s + 6·57-s − 14·59-s + 8·61-s − 14·67-s + 4·69-s − 10·73-s − 11·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 0.242·17-s + 1.37·19-s + 0.834·23-s − 0.962·27-s + 0.557·29-s − 0.359·31-s − 0.174·33-s − 1.31·37-s − 0.800·39-s + 1.56·41-s + 0.304·43-s − 1.02·47-s + 0.140·51-s + 0.274·53-s + 0.794·57-s − 1.82·59-s + 1.02·61-s − 1.71·67-s + 0.481·69-s − 1.17·73-s − 1.23·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| 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
−7.64775519989984714507232689938, −7.54198020453554071537282564567, −6.55042210418963463903437754298, −5.49542029150267177093826337208, −5.11652961027237543950690791152, −4.10431621261505539709645473683, −2.95308616915656117786688961081, −2.76219418784149403643777104301, −1.45796001870746498481271114091, 0,
1.45796001870746498481271114091, 2.76219418784149403643777104301, 2.95308616915656117786688961081, 4.10431621261505539709645473683, 5.11652961027237543950690791152, 5.49542029150267177093826337208, 6.55042210418963463903437754298, 7.54198020453554071537282564567, 7.64775519989984714507232689938
