| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 2·7-s − 8-s + 9-s + 2·12-s − 2·13-s + 2·14-s + 16-s − 6·17-s − 18-s + 2·19-s − 4·21-s − 2·24-s + 2·26-s − 4·27-s − 2·28-s + 6·29-s − 10·31-s − 32-s + 6·34-s + 36-s − 37-s − 2·38-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.872·21-s − 0.408·24-s + 0.392·26-s − 0.769·27-s − 0.377·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.164·37-s − 0.324·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−9.121399906792511920506761900085, −8.204020200703998161099385758919, −7.42242276633593108306611171999, −6.76983684462036733251522759049, −5.83762763428516435209739090188, −4.58625499065095739397162183000, −3.45045147360610483790234313369, −2.73896309603557156269653408686, −1.83232545260402930537453330354, 0,
1.83232545260402930537453330354, 2.73896309603557156269653408686, 3.45045147360610483790234313369, 4.58625499065095739397162183000, 5.83762763428516435209739090188, 6.76983684462036733251522759049, 7.42242276633593108306611171999, 8.204020200703998161099385758919, 9.121399906792511920506761900085
