| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 4·13-s − 2·14-s + 16-s + 17-s + 4·19-s + 20-s − 4·23-s + 25-s − 4·26-s + 2·28-s − 2·29-s − 32-s − 34-s + 2·35-s − 2·37-s − 4·38-s − 40-s + 4·41-s + 10·43-s + 4·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 0.371·29-s − 0.176·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.624·41-s + 1.52·43-s + 0.589·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.572004110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572004110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.353816570463711911551170941736, −8.744824833781760140637365800545, −7.893638261401034320528738567756, −7.30190083948422148597873963017, −6.12753069369136214734966964315, −5.60795183381495123088792753418, −4.40989647450811651870833634917, −3.28387977843809894542614546328, −2.04240923081670435558544642459, −1.06414014612462247343490013577,
1.06414014612462247343490013577, 2.04240923081670435558544642459, 3.28387977843809894542614546328, 4.40989647450811651870833634917, 5.60795183381495123088792753418, 6.12753069369136214734966964315, 7.30190083948422148597873963017, 7.893638261401034320528738567756, 8.744824833781760140637365800545, 9.353816570463711911551170941736
