| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 4·7-s − 8-s + 9-s + 10-s − 2·12-s + 4·14-s + 2·15-s + 16-s + 6·17-s − 18-s + 2·19-s − 20-s + 8·21-s + 6·23-s + 2·24-s + 25-s + 4·27-s − 4·28-s + 6·29-s − 2·30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 1.74·21-s + 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.755·28-s + 1.11·29-s − 0.365·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 204490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6690059825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6690059825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.75489487592543, −12.35578177486230, −12.03205001870631, −11.80671568804952, −11.02974833989965, −10.62899464028835, −10.35887845367477, −9.742340552664828, −9.432645612181142, −8.815982265345701, −8.376592335400694, −7.709292783505955, −7.163827624231259, −6.817939064505304, −6.404949455213182, −5.802925619140878, −5.398000411941062, −4.962417465335146, −4.098530073033655, −3.498696818435527, −2.981417344958052, −2.641964682432917, −1.432753097057366, −0.8761763168132313, −0.3813306853382419,
0.3813306853382419, 0.8761763168132313, 1.432753097057366, 2.641964682432917, 2.981417344958052, 3.498696818435527, 4.098530073033655, 4.962417465335146, 5.398000411941062, 5.802925619140878, 6.404949455213182, 6.817939064505304, 7.163827624231259, 7.709292783505955, 8.376592335400694, 8.815982265345701, 9.432645612181142, 9.742340552664828, 10.35887845367477, 10.62899464028835, 11.02974833989965, 11.80671568804952, 12.03205001870631, 12.35578177486230, 12.75489487592543
