| L(s) = 1 | − 3·5-s + 4·7-s − 3·11-s − 2·13-s + 8·19-s + 6·23-s + 4·25-s + 3·29-s + 7·31-s − 12·35-s − 8·37-s − 6·41-s − 4·43-s − 6·47-s + 9·49-s − 9·53-s + 9·55-s − 15·59-s − 14·61-s + 6·65-s + 2·67-s − 7·73-s − 12·77-s + 79-s − 12·83-s − 8·91-s − 24·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s − 0.904·11-s − 0.554·13-s + 1.83·19-s + 1.25·23-s + 4/5·25-s + 0.557·29-s + 1.25·31-s − 2.02·35-s − 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s − 1.23·53-s + 1.21·55-s − 1.95·59-s − 1.79·61-s + 0.744·65-s + 0.244·67-s − 0.819·73-s − 1.36·77-s + 0.112·79-s − 1.31·83-s − 0.838·91-s − 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.194679296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.194679296\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36978400639748, −12.56912256842974, −12.04515633859823, −11.94101758768707, −11.30395486968028, −11.07721119746315, −10.47165737409391, −10.02749646774307, −9.365135351468291, −8.773376228795720, −8.241318860901584, −7.900821894135887, −7.587738352492568, −7.121531508682600, −6.575630160995466, −5.675765716075045, −5.009891396855983, −4.878378462691267, −4.504229462736921, −3.620252287169588, −2.985361787462851, −2.815374928699214, −1.575329062089173, −1.334071311901253, −0.3273726628514030,
0.3273726628514030, 1.334071311901253, 1.575329062089173, 2.815374928699214, 2.985361787462851, 3.620252287169588, 4.504229462736921, 4.878378462691267, 5.009891396855983, 5.675765716075045, 6.575630160995466, 7.121531508682600, 7.587738352492568, 7.900821894135887, 8.241318860901584, 8.773376228795720, 9.365135351468291, 10.02749646774307, 10.47165737409391, 11.07721119746315, 11.30395486968028, 11.94101758768707, 12.04515633859823, 12.56912256842974, 13.36978400639748
