| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 4·7-s + 8-s + 9-s − 10-s − 12-s − 2·13-s + 4·14-s + 15-s + 16-s − 2·17-s + 18-s − 20-s − 4·21-s − 24-s + 25-s − 2·26-s − 27-s + 4·28-s − 10·29-s + 30-s + 32-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s + 0.182·30-s + 0.176·32-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−16.77332486404449, −16.29471690493812, −15.37326995975685, −15.15492306139839, −14.57977897985406, −14.04952183076417, −13.30659642060461, −12.80407380725367, −12.05909740338042, −11.58984932506059, −11.26327908308439, −10.65146077930927, −10.03820199353102, −9.072835002102555, −8.447042645459931, −7.622548017201207, −7.332299876052681, −6.556863639180583, −5.592756137155789, −5.269700195218328, −4.456553796644732, −4.119196133866012, −3.078259935823343, −2.045346834982421, −1.417299444498659, 0,
1.417299444498659, 2.045346834982421, 3.078259935823343, 4.119196133866012, 4.456553796644732, 5.269700195218328, 5.592756137155789, 6.556863639180583, 7.332299876052681, 7.622548017201207, 8.447042645459931, 9.072835002102555, 10.03820199353102, 10.65146077930927, 11.26327908308439, 11.58984932506059, 12.05909740338042, 12.80407380725367, 13.30659642060461, 14.04952183076417, 14.57977897985406, 15.15492306139839, 15.37326995975685, 16.29471690493812, 16.77332486404449
