| L(s) = 1 | − 4·5-s − 2·7-s − 4·11-s + 2·13-s + 4·17-s + 8·23-s + 11·25-s − 6·29-s + 8·35-s + 10·37-s − 8·41-s − 8·43-s − 3·49-s + 8·53-s + 16·55-s + 8·59-s − 10·61-s − 8·65-s − 12·67-s − 12·71-s + 10·73-s + 8·77-s − 12·79-s − 4·83-s − 16·85-s − 4·91-s + 14·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s + 0.970·17-s + 1.66·23-s + 11/5·25-s − 1.11·29-s + 1.35·35-s + 1.64·37-s − 1.24·41-s − 1.21·43-s − 3/7·49-s + 1.09·53-s + 2.15·55-s + 1.04·59-s − 1.28·61-s − 0.992·65-s − 1.46·67-s − 1.42·71-s + 1.17·73-s + 0.911·77-s − 1.35·79-s − 0.439·83-s − 1.73·85-s − 0.419·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8123011258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8123011258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−12.87800746305881, −12.55443113789793, −11.89615441579235, −11.59806311729519, −11.12008419509985, −10.71410697516059, −10.19665600808384, −9.749114394946487, −9.061092440075373, −8.620686708278508, −8.185217829305474, −7.688770781021720, −7.300914545687011, −6.971582882040246, −6.257352520278619, −5.681921966052095, −5.114301610876918, −4.663844827409737, −4.045788921459703, −3.494190904038725, −3.039607543814554, −2.840922965674616, −1.711455807122220, −0.9127010280833380, −0.3136881268687598,
0.3136881268687598, 0.9127010280833380, 1.711455807122220, 2.840922965674616, 3.039607543814554, 3.494190904038725, 4.045788921459703, 4.663844827409737, 5.114301610876918, 5.681921966052095, 6.257352520278619, 6.971582882040246, 7.300914545687011, 7.688770781021720, 8.185217829305474, 8.620686708278508, 9.061092440075373, 9.749114394946487, 10.19665600808384, 10.71410697516059, 11.12008419509985, 11.59806311729519, 11.89615441579235, 12.55443113789793, 12.87800746305881
