| L(s) = 1 | − 2·5-s − 3·9-s − 4·11-s + 4·13-s + 8·17-s + 2·19-s − 23-s − 25-s − 2·29-s − 6·31-s + 10·37-s − 6·41-s − 8·43-s + 6·45-s + 6·47-s − 2·53-s + 8·55-s + 10·61-s − 8·65-s + 8·67-s + 12·71-s − 6·73-s + 9·81-s − 2·83-s − 16·85-s − 12·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.20·11-s + 1.10·13-s + 1.94·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.07·31-s + 1.64·37-s − 0.937·41-s − 1.21·43-s + 0.894·45-s + 0.875·47-s − 0.274·53-s + 1.07·55-s + 1.28·61-s − 0.992·65-s + 0.977·67-s + 1.42·71-s − 0.702·73-s + 81-s − 0.219·83-s − 1.73·85-s − 1.27·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
| 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
−14.41251846696305, −13.82119550093230, −13.40537458867393, −12.84312048049550, −12.28824466816244, −11.82959425481854, −11.30980184547427, −11.00473852104440, −10.41660652227859, −9.752915258506114, −9.408062123348308, −8.448560419822519, −8.186582793147304, −7.878305389987010, −7.307091641101418, −6.619204655629843, −5.806522822204902, −5.471120899981748, −5.136571286505106, −4.039548782623374, −3.686651123093911, −3.103951321514964, −2.589069743562643, −1.602172163424589, −0.7828947544105197, 0,
0.7828947544105197, 1.602172163424589, 2.589069743562643, 3.103951321514964, 3.686651123093911, 4.039548782623374, 5.136571286505106, 5.471120899981748, 5.806522822204902, 6.619204655629843, 7.307091641101418, 7.878305389987010, 8.186582793147304, 8.448560419822519, 9.408062123348308, 9.752915258506114, 10.41660652227859, 11.00473852104440, 11.30980184547427, 11.82959425481854, 12.28824466816244, 12.84312048049550, 13.40537458867393, 13.82119550093230, 14.41251846696305
