| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·11-s − 2·13-s − 15-s + 8·17-s − 2·19-s − 2·21-s + 6·23-s + 25-s − 27-s + 6·29-s + 2·33-s + 2·35-s − 10·37-s + 2·39-s + 2·41-s + 43-s + 45-s + 6·47-s − 3·49-s − 8·51-s − 2·53-s − 2·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s + 1.94·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.348·33-s + 0.338·35-s − 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 1.12·51-s − 0.274·53-s − 0.269·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492450720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492450720\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.74520935138928, −14.23000051082432, −13.82497121531958, −13.09581199956597, −12.56521615049352, −12.21024108944224, −11.65447157077066, −11.07074992087718, −10.40460342536298, −10.24698411357922, −9.615387282443483, −8.866287384181611, −8.358393158428959, −7.715418596189541, −7.234899869575452, −6.690336467406376, −5.855875034149000, −5.435566607133047, −4.949587953514400, −4.484299274882506, −3.475738165779368, −2.897660217614926, −2.085028673183570, −1.321756191459917, −0.6412987206949875,
0.6412987206949875, 1.321756191459917, 2.085028673183570, 2.897660217614926, 3.475738165779368, 4.484299274882506, 4.949587953514400, 5.435566607133047, 5.855875034149000, 6.690336467406376, 7.234899869575452, 7.715418596189541, 8.358393158428959, 8.866287384181611, 9.615387282443483, 10.24698411357922, 10.40460342536298, 11.07074992087718, 11.65447157077066, 12.21024108944224, 12.56521615049352, 13.09581199956597, 13.82497121531958, 14.23000051082432, 14.74520935138928
