| L(s) = 1 | − 5-s + 2·11-s − 6·13-s − 8·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 4·31-s + 2·37-s − 4·41-s − 6·43-s + 8·47-s + 2·53-s − 2·55-s + 4·59-s − 10·61-s + 6·65-s + 2·67-s − 8·71-s − 12·73-s + 4·79-s − 12·83-s + 8·85-s − 16·89-s − 4·95-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 1.66·13-s − 1.94·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.624·41-s − 0.914·43-s + 1.16·47-s + 0.274·53-s − 0.269·55-s + 0.520·59-s − 1.28·61-s + 0.744·65-s + 0.244·67-s − 0.949·71-s − 1.40·73-s + 0.450·79-s − 1.31·83-s + 0.867·85-s − 1.69·89-s − 0.410·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 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.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−12.86168081362315, −12.48427663678433, −11.86940768972194, −11.79415381113779, −11.19291648127910, −10.76147349737051, −10.07510411297570, −9.824184727147760, −9.281044033182516, −8.734809395575282, −8.498964654962336, −7.706157409494488, −7.309723035985523, −6.993996894270072, −6.492058125871103, −5.856807468520635, −5.336111828201634, −4.722008693611233, −4.329448884400371, −3.976378309326254, −3.106462974544129, −2.746308202679515, −2.019047999510468, −1.605368994186198, −0.5737039694904942, 0,
0.5737039694904942, 1.605368994186198, 2.019047999510468, 2.746308202679515, 3.106462974544129, 3.976378309326254, 4.329448884400371, 4.722008693611233, 5.336111828201634, 5.856807468520635, 6.492058125871103, 6.993996894270072, 7.309723035985523, 7.706157409494488, 8.498964654962336, 8.734809395575282, 9.281044033182516, 9.824184727147760, 10.07510411297570, 10.76147349737051, 11.19291648127910, 11.79415381113779, 11.86940768972194, 12.48427663678433, 12.86168081362315
