| L(s) = 1 | + 3-s + 4·5-s + 9-s − 5·13-s + 4·15-s + 8·17-s + 4·19-s − 6·23-s + 11·25-s + 27-s + 31-s + 5·37-s − 5·39-s + 6·41-s + 9·43-s + 4·45-s − 2·47-s + 8·51-s + 8·53-s + 4·57-s − 12·59-s − 11·61-s − 20·65-s + 8·67-s − 6·69-s − 12·71-s + 9·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.38·13-s + 1.03·15-s + 1.94·17-s + 0.917·19-s − 1.25·23-s + 11/5·25-s + 0.192·27-s + 0.179·31-s + 0.821·37-s − 0.800·39-s + 0.937·41-s + 1.37·43-s + 0.596·45-s − 0.291·47-s + 1.12·51-s + 1.09·53-s + 0.529·57-s − 1.56·59-s − 1.40·61-s − 2.48·65-s + 0.977·67-s − 0.722·69-s − 1.42·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.153184754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.153184754\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.66739268497755, −12.87279418144164, −12.50802555576603, −12.14907995279613, −11.61773290010475, −10.73609859012357, −10.33446321387942, −9.770061677127855, −9.712131726585579, −9.267208870691894, −8.666768909171023, −7.876310498392619, −7.522908628512589, −7.228339175104156, −6.191310400847618, −6.012525532776888, −5.501824705182538, −4.930491206855138, −4.442704145522921, −3.554868110153868, −2.985459136415918, −2.484304190537427, −2.018812392724242, −1.320933497020985, −0.7263568604449006,
0.7263568604449006, 1.320933497020985, 2.018812392724242, 2.484304190537427, 2.985459136415918, 3.554868110153868, 4.442704145522921, 4.930491206855138, 5.501824705182538, 6.012525532776888, 6.191310400847618, 7.228339175104156, 7.522908628512589, 7.876310498392619, 8.666768909171023, 9.267208870691894, 9.712131726585579, 9.770061677127855, 10.33446321387942, 10.73609859012357, 11.61773290010475, 12.14907995279613, 12.50802555576603, 12.87279418144164, 13.66739268497755
