| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 2·11-s − 4·13-s − 15-s + 2·17-s − 2·19-s + 21-s − 4·23-s + 25-s + 27-s + 2·29-s + 6·31-s + 2·33-s − 35-s + 6·37-s − 4·39-s + 6·41-s − 4·43-s − 45-s + 49-s + 2·51-s − 8·53-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.348·33-s − 0.169·35-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.280·51-s − 1.09·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340298000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340298000\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−8.049079847482147943302556686432, −7.43219226405919851517472847554, −6.67714151309704517931518407289, −5.96260516862523232882973830545, −4.89501828327550470087725504492, −4.39088989617993719988896336014, −3.61002146901025333628515331519, −2.71503969140231506326507050220, −1.94218971622800952578247712625, −0.76554310889382658952076086492,
0.76554310889382658952076086492, 1.94218971622800952578247712625, 2.71503969140231506326507050220, 3.61002146901025333628515331519, 4.39088989617993719988896336014, 4.89501828327550470087725504492, 5.96260516862523232882973830545, 6.67714151309704517931518407289, 7.43219226405919851517472847554, 8.049079847482147943302556686432
