| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 11-s − 15-s + 2·17-s − 6·19-s − 2·21-s + 3·23-s + 25-s + 27-s − 29-s + 3·31-s − 33-s + 2·35-s − 5·37-s + 10·41-s − 5·43-s − 45-s − 3·47-s − 3·49-s + 2·51-s + 14·53-s + 55-s − 6·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.174·33-s + 0.338·35-s − 0.821·37-s + 1.56·41-s − 0.762·43-s − 0.149·45-s − 0.437·47-s − 3/7·49-s + 0.280·51-s + 1.92·53-s + 0.134·55-s − 0.794·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−15.11006095087128, −14.55633852207573, −14.05073403150268, −13.32708223340180, −13.00812626771830, −12.56486186443929, −11.98378339174837, −11.40724176419647, −10.70141764236213, −10.28670569015093, −9.811617292900838, −9.032881471775070, −8.748368147442645, −8.115567782998181, −7.543247284478837, −7.020166321272706, −6.417017338042200, −5.877367463082931, −5.054917025089402, −4.441995381044943, −3.804445358087738, −3.241153907225774, −2.637478061833213, −1.942584157191138, −0.9292540090669781, 0,
0.9292540090669781, 1.942584157191138, 2.637478061833213, 3.241153907225774, 3.804445358087738, 4.441995381044943, 5.054917025089402, 5.877367463082931, 6.417017338042200, 7.020166321272706, 7.543247284478837, 8.115567782998181, 8.748368147442645, 9.032881471775070, 9.811617292900838, 10.28670569015093, 10.70141764236213, 11.40724176419647, 11.98378339174837, 12.56486186443929, 13.00812626771830, 13.32708223340180, 14.05073403150268, 14.55633852207573, 15.11006095087128
