| L(s) = 1 | + 2·3-s + 3·5-s + 4·7-s + 9-s + 6·11-s + 2·13-s + 6·15-s + 3·17-s − 2·19-s + 8·21-s − 6·23-s + 4·25-s − 4·27-s + 3·29-s − 2·31-s + 12·33-s + 12·35-s + 4·39-s + 3·41-s + 4·43-s + 3·45-s + 6·47-s + 9·49-s + 6·51-s − 6·53-s + 18·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 1.54·15-s + 0.727·17-s − 0.458·19-s + 1.74·21-s − 1.25·23-s + 4/5·25-s − 0.769·27-s + 0.557·29-s − 0.359·31-s + 2.08·33-s + 2.02·35-s + 0.640·39-s + 0.468·41-s + 0.609·43-s + 0.447·45-s + 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 2.42·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.190179459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.190179459\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
| 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
−15.31157663346198, −14.63890651824463, −14.27226098084260, −14.15841362642864, −13.73821500314859, −12.99967223477918, −12.26417846899733, −11.72243635755504, −11.16914769229914, −10.51615262701303, −9.813131214945725, −9.382588737163377, −8.791768719125714, −8.450275098634812, −7.823859273076310, −7.188954637901711, −6.253874165364370, −5.932076259301143, −5.225674938068710, −4.229416320415736, −3.955535368666894, −2.976157486553858, −2.161402156529859, −1.658858866713202, −1.168859188211210,
1.168859188211210, 1.658858866713202, 2.161402156529859, 2.976157486553858, 3.955535368666894, 4.229416320415736, 5.225674938068710, 5.932076259301143, 6.253874165364370, 7.188954637901711, 7.823859273076310, 8.450275098634812, 8.791768719125714, 9.382588737163377, 9.813131214945725, 10.51615262701303, 11.16914769229914, 11.72243635755504, 12.26417846899733, 12.99967223477918, 13.73821500314859, 14.15841362642864, 14.27226098084260, 14.63890651824463, 15.31157663346198
