| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 2·14-s + 15-s + 16-s + 2·17-s − 18-s + 6·19-s − 20-s + 2·21-s − 22-s + 8·23-s + 24-s + 25-s − 27-s − 2·28-s − 8·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s − 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 89 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.61243227744403, −15.06143421382243, −14.54880430250095, −13.71522171116075, −13.28586753857175, −12.54017759101863, −12.20462227309062, −11.55313498943706, −11.21582378216552, −10.56069010507606, −10.01317436463689, −9.467000748584421, −9.054136480989472, −8.375437749435876, −7.648608539069047, −7.082120700986386, −6.821573587734998, −6.028995553904453, −5.326080359028891, −4.938652819961728, −3.774001868582894, −3.400482648742266, −2.680101110882059, −1.553363440395729, −0.8971919297693536, 0,
0.8971919297693536, 1.553363440395729, 2.680101110882059, 3.400482648742266, 3.774001868582894, 4.938652819961728, 5.326080359028891, 6.028995553904453, 6.821573587734998, 7.082120700986386, 7.648608539069047, 8.375437749435876, 9.054136480989472, 9.467000748584421, 10.01317436463689, 10.56069010507606, 11.21582378216552, 11.55313498943706, 12.20462227309062, 12.54017759101863, 13.28586753857175, 13.71522171116075, 14.54880430250095, 15.06143421382243, 15.61243227744403
