| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 4·7-s − 3·8-s + 9-s + 12-s − 4·14-s − 16-s + 18-s + 19-s + 4·21-s + 4·23-s + 3·24-s − 27-s + 4·28-s + 8·29-s − 4·31-s + 5·32-s − 36-s + 2·37-s + 38-s + 2·41-s + 4·42-s − 2·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.06·14-s − 1/4·16-s + 0.235·18-s + 0.229·19-s + 0.872·21-s + 0.834·23-s + 0.612·24-s − 0.192·27-s + 0.755·28-s + 1.48·29-s − 0.718·31-s + 0.883·32-s − 1/6·36-s + 0.328·37-s + 0.162·38-s + 0.312·41-s + 0.617·42-s − 0.304·43-s + 0.589·46-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
| \(L(1)\) |
\(\approx\) |
\(1.557006181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557006181\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.91591760329520, −12.46121930645739, −12.16929367065961, −11.71692226567073, −11.00542898908155, −10.64833094839448, −10.04985084780797, −9.554149889769449, −9.352749286179746, −8.800244839154040, −8.207376722765469, −7.629484583364742, −6.937094269720819, −6.449544204436123, −6.323985396591045, −5.579598059543608, −5.228418213737644, −4.670872000806004, −4.155531248397492, −3.614181414388869, −3.013093311756531, −2.829447999428282, −1.818457615054543, −0.8561276361499729, −0.4166590050397929,
0.4166590050397929, 0.8561276361499729, 1.818457615054543, 2.829447999428282, 3.013093311756531, 3.614181414388869, 4.155531248397492, 4.670872000806004, 5.228418213737644, 5.579598059543608, 6.323985396591045, 6.449544204436123, 6.937094269720819, 7.629484583364742, 8.207376722765469, 8.800244839154040, 9.352749286179746, 9.554149889769449, 10.04985084780797, 10.64833094839448, 11.00542898908155, 11.71692226567073, 12.16929367065961, 12.46121930645739, 12.91591760329520
