| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 12-s − 16-s + 6·17-s + 18-s + 19-s − 4·23-s + 3·24-s − 27-s + 2·29-s − 8·31-s + 5·32-s + 6·34-s − 36-s − 10·37-s + 38-s + 2·41-s + 4·43-s − 4·46-s + 12·47-s + 48-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.834·23-s + 0.612·24-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.883·32-s + 1.02·34-s − 1/6·36-s − 1.64·37-s + 0.162·38-s + 0.312·41-s + 0.609·43-s − 0.589·46-s + 1.75·47-s + 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153842868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153842868\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82426215498061, −12.35775249838551, −12.07645124232765, −11.79729280473272, −11.08032932325141, −10.50205783425754, −10.26095561030236, −9.569648953827147, −9.281527590767685, −8.719028193786285, −8.117848544007903, −7.706088469471329, −7.088685043096862, −6.643393806444784, −5.923729621810463, −5.524173613687345, −5.371279101869549, −4.696578163504022, −4.113887596302248, −3.609949454524708, −3.323977980996247, −2.467590137964769, −1.832123220451570, −0.9830974936223205, −0.4443127558399791,
0.4443127558399791, 0.9830974936223205, 1.832123220451570, 2.467590137964769, 3.323977980996247, 3.609949454524708, 4.113887596302248, 4.696578163504022, 5.371279101869549, 5.524173613687345, 5.923729621810463, 6.643393806444784, 7.088685043096862, 7.706088469471329, 8.117848544007903, 8.719028193786285, 9.281527590767685, 9.569648953827147, 10.26095561030236, 10.50205783425754, 11.08032932325141, 11.79729280473272, 12.07645124232765, 12.35775249838551, 12.82426215498061
