| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2·11-s + 12-s − 4·13-s + 16-s − 2·17-s − 18-s + 2·22-s + 8·23-s − 24-s + 4·26-s + 27-s − 6·29-s − 2·31-s − 32-s − 2·33-s + 2·34-s + 36-s + 2·37-s − 4·39-s + 10·41-s + 12·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.426·22-s + 1.66·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.640·39-s + 1.56·41-s + 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 61 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.54433064901946554429629952868, −6.99963379462494163973012130842, −6.16919000031421177672010997801, −5.27471739825455170417996911844, −4.64456815723790908416860205038, −3.68287729117018277941373693983, −2.72883720322617491987483404159, −2.32871530521889101014496037067, −1.20836580994259274155184079881, 0,
1.20836580994259274155184079881, 2.32871530521889101014496037067, 2.72883720322617491987483404159, 3.68287729117018277941373693983, 4.64456815723790908416860205038, 5.27471739825455170417996911844, 6.16919000031421177672010997801, 6.99963379462494163973012130842, 7.54433064901946554429629952868
