| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s − 12-s + 4·13-s + 14-s − 2·15-s + 16-s − 18-s + 2·20-s + 21-s − 6·23-s + 24-s − 25-s − 4·26-s − 27-s − 28-s + 4·29-s + 2·30-s − 32-s − 2·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.10·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s + 0.365·30-s − 0.176·32-s − 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−16.59897399005509, −16.06247778514387, −15.95035397135932, −15.04013871593707, −14.39454524269330, −13.61894848461225, −13.36006802069396, −12.56252142347512, −11.98609907142957, −11.39916320860981, −10.73686041781557, −10.26238603182298, −9.708328285118823, −9.246423360541617, −8.427415062070016, −7.960475951391367, −7.046397841865952, −6.412034380830504, −5.983331791329172, −5.483483830821573, −4.459321027449694, −3.674091294608471, −2.755294859619879, −1.854898700624538, −1.171979980125002, 0,
1.171979980125002, 1.854898700624538, 2.755294859619879, 3.674091294608471, 4.459321027449694, 5.483483830821573, 5.983331791329172, 6.412034380830504, 7.046397841865952, 7.960475951391367, 8.427415062070016, 9.246423360541617, 9.708328285118823, 10.26238603182298, 10.73686041781557, 11.39916320860981, 11.98609907142957, 12.56252142347512, 13.36006802069396, 13.61894848461225, 14.39454524269330, 15.04013871593707, 15.95035397135932, 16.06247778514387, 16.59897399005509
