| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 3·7-s + 3·8-s + 9-s − 10-s − 5·11-s − 12-s + 3·14-s + 15-s − 16-s − 2·17-s − 18-s − 6·19-s − 20-s − 3·21-s + 5·22-s − 2·23-s + 3·24-s + 25-s + 27-s + 3·28-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 − 1.13·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.801·14-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 − 0.654·21-s + 1.06·22-s − 0.417·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.566·28-s − 1.48·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−15.83847155818299, −15.28638947746649, −14.67253312976131, −14.20049904678468, −13.34903036125633, −13.15668486766153, −12.86419623692188, −12.33364094310958, −11.06233392947143, −10.70647797777841, −10.34703876818457, −9.572973340025002, −9.256361982987644, −8.900952959612990, −8.115114676218915, −7.568996242450002, −7.245197050982058, −6.181899295263499, −5.859430053436573, −4.987061935779069, −4.315594344193848, −3.703420320620561, −2.870979438054664, −2.238563997490182, −1.538745466362316, 0, 0,
1.538745466362316, 2.238563997490182, 2.870979438054664, 3.703420320620561, 4.315594344193848, 4.987061935779069, 5.859430053436573, 6.181899295263499, 7.245197050982058, 7.568996242450002, 8.115114676218915, 8.900952959612990, 9.256361982987644, 9.572973340025002, 10.34703876818457, 10.70647797777841, 11.06233392947143, 12.33364094310958, 12.86419623692188, 13.15668486766153, 13.34903036125633, 14.20049904678468, 14.67253312976131, 15.28638947746649, 15.83847155818299
