| L(s) = 1 | − 2·5-s − 3·9-s − 6·13-s + 2·17-s − 25-s + 10·29-s + 2·37-s − 10·41-s + 6·45-s − 7·49-s − 14·53-s − 10·61-s + 12·65-s − 6·73-s + 9·81-s − 4·85-s − 10·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.66·13-s + 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s − 1.56·41-s + 0.894·45-s − 49-s − 1.92·53-s − 1.28·61-s + 1.48·65-s − 0.702·73-s + 81-s − 0.433·85-s − 1.05·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6015412580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6015412580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
| 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
−16.57361893192400, −15.77225634783173, −15.29628913235154, −14.74469448018006, −14.18076451594602, −13.81810451385395, −12.83260616576279, −12.23993141789085, −11.90203799626850, −11.40992992227961, −10.67892058394557, −9.987259191289484, −9.504189729118573, −8.668231811646329, −8.035633245388709, −7.736309920139672, −6.882856708802471, −6.307162268715449, −5.379922843906619, −4.829219040935091, −4.203567179262050, −3.088341052646371, −2.870145436760538, −1.686503601074422, −0.3438831671720621,
0.3438831671720621, 1.686503601074422, 2.870145436760538, 3.088341052646371, 4.203567179262050, 4.829219040935091, 5.379922843906619, 6.307162268715449, 6.882856708802471, 7.736309920139672, 8.035633245388709, 8.668231811646329, 9.504189729118573, 9.987259191289484, 10.67892058394557, 11.40992992227961, 11.90203799626850, 12.23993141789085, 12.83260616576279, 13.81810451385395, 14.18076451594602, 14.74469448018006, 15.29628913235154, 15.77225634783173, 16.57361893192400
