| L(s) = 1 | + 3·3-s − 2·4-s − 2·5-s − 7-s + 6·9-s + 3·11-s − 6·12-s + 6·13-s − 6·15-s + 4·16-s + 2·17-s + 19-s + 4·20-s − 3·21-s − 25-s + 9·27-s + 2·28-s + 6·29-s − 4·31-s + 9·33-s + 2·35-s − 12·36-s − 37-s + 18·39-s + 9·41-s + 4·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s − 0.894·5-s − 0.377·7-s + 2·9-s + 0.904·11-s − 1.73·12-s + 1.66·13-s − 1.54·15-s + 16-s + 0.485·17-s + 0.229·19-s + 0.894·20-s − 0.654·21-s − 1/5·25-s + 1.73·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 1.56·33-s + 0.338·35-s − 2·36-s − 0.164·37-s + 2.88·39-s + 1.40·41-s + 0.609·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053106134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053106134\) |
| \(L(\frac{3}{2})\) |
|
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 | 19 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−10.05447016331795076753843124558, −9.322785243387073018175081080931, −8.649334900934084695834907889147, −8.172596918335991860172932097060, −7.32057820339219969391229332885, −6.06731610306162376296883739999, −4.41700371545169852157216919625, −3.73679275260902051694693414558, −3.19093175218016983244031724621, −1.29286312905799335016720966581,
1.29286312905799335016720966581, 3.19093175218016983244031724621, 3.73679275260902051694693414558, 4.41700371545169852157216919625, 6.06731610306162376296883739999, 7.32057820339219969391229332885, 8.172596918335991860172932097060, 8.649334900934084695834907889147, 9.322785243387073018175081080931, 10.05447016331795076753843124558
