| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 3·11-s − 12-s − 13-s − 15-s + 16-s − 5·17-s + 18-s + 19-s + 20-s − 3·22-s − 4·23-s − 24-s + 25-s − 26-s − 27-s − 8·29-s − 30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.639·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 1.48·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.97757727681340, −12.23513009313471, −11.87825111214465, −11.51392604049901, −10.90525459919632, −10.59646662330702, −10.13492373205405, −9.790316543862577, −9.068935285994866, −8.707890107627382, −8.070224109450812, −7.545185823129516, −7.075894230020282, −6.706767828041349, −6.144829658356963, −5.521734901300990, −5.456887393941508, −4.818906581798011, −4.288902848113855, −3.847210673433558, −3.219148432210488, −2.464399554845752, −2.168341529086906, −1.613099933822173, −0.6881260777454889, 0,
0.6881260777454889, 1.613099933822173, 2.168341529086906, 2.464399554845752, 3.219148432210488, 3.847210673433558, 4.288902848113855, 4.818906581798011, 5.456887393941508, 5.521734901300990, 6.144829658356963, 6.706767828041349, 7.075894230020282, 7.545185823129516, 8.070224109450812, 8.707890107627382, 9.068935285994866, 9.790316543862577, 10.13492373205405, 10.59646662330702, 10.90525459919632, 11.51392604049901, 11.87825111214465, 12.23513009313471, 12.97757727681340
