| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 3·11-s − 2·13-s − 4·14-s − 16-s − 3·17-s − 6·19-s − 3·22-s − 4·23-s − 2·26-s + 4·28-s − 6·29-s − 3·31-s + 5·32-s − 3·34-s − 6·37-s − 6·38-s − 41-s − 5·43-s + 3·44-s − 4·46-s + 7·47-s + 9·49-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 0.904·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.639·22-s − 0.834·23-s − 0.392·26-s + 0.755·28-s − 1.11·29-s − 0.538·31-s + 0.883·32-s − 0.514·34-s − 0.986·37-s − 0.973·38-s − 0.156·41-s − 0.762·43-s + 0.452·44-s − 0.589·46-s + 1.02·47-s + 9/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−6.84453434348862534645963191293, −6.21991349221845952452333751657, −5.66229658075253023147174372152, −4.93114770470381338534535138791, −4.14825817067842689970052025400, −3.57867319204935335138993074405, −2.80057586577332264564523382934, −2.06037232312361280137635877425, 0, 0,
2.06037232312361280137635877425, 2.80057586577332264564523382934, 3.57867319204935335138993074405, 4.14825817067842689970052025400, 4.93114770470381338534535138791, 5.66229658075253023147174372152, 6.21991349221845952452333751657, 6.84453434348862534645963191293
