| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·11-s + 13-s + 15-s − 7·17-s − 5·19-s + 2·21-s − 6·23-s − 4·25-s − 27-s + 7·31-s + 2·33-s + 2·35-s + 2·37-s − 39-s + 41-s − 4·43-s − 45-s − 12·47-s − 3·49-s + 7·51-s + 6·53-s + 2·55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.258·15-s − 1.69·17-s − 1.14·19-s + 0.436·21-s − 1.25·23-s − 4/5·25-s − 0.192·27-s + 1.25·31-s + 0.348·33-s + 0.338·35-s + 0.328·37-s − 0.160·39-s + 0.156·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.980·51-s + 0.824·53-s + 0.269·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3058825092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3058825092\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−7.948880214801955346466380200848, −6.99228337894628662827672485486, −6.30911952261537959207628464072, −6.08402989133586573489513657933, −4.91625326570346480713950345318, −4.34399659988896875970045777032, −3.67019757439157696988330467218, −2.63262293864378919969029811262, −1.82289196395902310303156227024, −0.26747535629309804136023280389,
0.26747535629309804136023280389, 1.82289196395902310303156227024, 2.63262293864378919969029811262, 3.67019757439157696988330467218, 4.34399659988896875970045777032, 4.91625326570346480713950345318, 6.08402989133586573489513657933, 6.30911952261537959207628464072, 6.99228337894628662827672485486, 7.948880214801955346466380200848
