| L(s) = 1 | + 3-s + 9-s + 13-s − 4·19-s − 5·25-s + 27-s + 6·29-s + 10·31-s + 4·37-s + 39-s + 6·41-s − 10·43-s + 2·47-s − 7·49-s + 14·53-s − 4·57-s − 2·61-s + 8·67-s − 6·71-s − 10·73-s − 5·75-s + 8·79-s + 81-s + 8·83-s + 6·87-s + 6·89-s + 10·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.917·19-s − 25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.657·37-s + 0.160·39-s + 0.937·41-s − 1.52·43-s + 0.291·47-s − 49-s + 1.92·53-s − 0.529·57-s − 0.256·61-s + 0.977·67-s − 0.712·71-s − 1.17·73-s − 0.577·75-s + 0.900·79-s + 1/9·81-s + 0.878·83-s + 0.643·87-s + 0.635·89-s + 1.03·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.350457549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.350457549\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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.ak |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.36639712306043, −13.02100851996683, −12.38879038634420, −11.82654353903299, −11.61510112202782, −10.87699863067817, −10.31222959975356, −10.05038066582675, −9.522802795968476, −8.872062405248204, −8.503911792491853, −8.035831860119711, −7.658374309115031, −6.925904224633466, −6.401349434050967, −6.118226245077648, −5.358906849882531, −4.657180359153792, −4.321993043863153, −3.709437343321722, −3.097837553465306, −2.489983560270632, −2.030235596574436, −1.209698476376912, −0.5449485207898809,
0.5449485207898809, 1.209698476376912, 2.030235596574436, 2.489983560270632, 3.097837553465306, 3.709437343321722, 4.321993043863153, 4.657180359153792, 5.358906849882531, 6.118226245077648, 6.401349434050967, 6.925904224633466, 7.658374309115031, 8.035831860119711, 8.503911792491853, 8.872062405248204, 9.522802795968476, 10.05038066582675, 10.31222959975356, 10.87699863067817, 11.61510112202782, 11.82654353903299, 12.38879038634420, 13.02100851996683, 13.36639712306043
