| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 13-s − 2·15-s + 6·17-s − 4·19-s + 21-s − 8·23-s − 25-s − 27-s − 6·29-s + 4·33-s − 2·35-s + 10·37-s − 39-s − 10·41-s + 8·43-s + 2·45-s + 49-s − 6·51-s + 10·53-s − 8·55-s + 4·57-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.696·33-s − 0.338·35-s + 1.64·37-s − 0.160·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s + 1/7·49-s − 0.840·51-s + 1.37·53-s − 1.07·55-s + 0.529·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.415428540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415428540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−7.913411912852685135332806341433, −6.97942625921481423551283451383, −6.14822347148578719625694780135, −5.72621381166205778106637759574, −5.30104558453416413718278589758, −4.25558382054487596934541958955, −3.52767668627418282399123296987, −2.45777446352643157868367979333, −1.83963101327050607475066311503, −0.57937443109427047835419380135,
0.57937443109427047835419380135, 1.83963101327050607475066311503, 2.45777446352643157868367979333, 3.52767668627418282399123296987, 4.25558382054487596934541958955, 5.30104558453416413718278589758, 5.72621381166205778106637759574, 6.14822347148578719625694780135, 6.97942625921481423551283451383, 7.913411912852685135332806341433
