| L(s) = 1 | − 2·5-s − 2·7-s − 4·11-s + 13-s + 6·19-s − 4·23-s − 25-s − 8·29-s − 2·31-s + 4·35-s − 6·37-s − 6·41-s + 8·43-s − 8·47-s − 3·49-s + 12·53-s + 8·55-s + 4·59-s − 10·61-s − 2·65-s + 2·67-s + 16·71-s + 14·73-s + 8·77-s − 4·79-s − 12·83-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s + 1.37·19-s − 0.834·23-s − 1/5·25-s − 1.48·29-s − 0.359·31-s + 0.676·35-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.64·53-s + 1.07·55-s + 0.520·59-s − 1.28·61-s − 0.248·65-s + 0.244·67-s + 1.89·71-s + 1.63·73-s + 0.911·77-s − 0.450·79-s − 1.31·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6913264760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6913264760\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.925372306604668271806604899129, −7.26580818596079410743963968531, −6.65703429532393217149424074574, −5.54509353956096526188985932549, −5.33096126389198272030573476032, −4.11034461023913425350748189621, −3.56316576941236052872728412075, −2.88405431956086540351190972945, −1.82245185313753041575480828826, −0.40028421621281780497383342011,
0.40028421621281780497383342011, 1.82245185313753041575480828826, 2.88405431956086540351190972945, 3.56316576941236052872728412075, 4.11034461023913425350748189621, 5.33096126389198272030573476032, 5.54509353956096526188985932549, 6.65703429532393217149424074574, 7.26580818596079410743963968531, 7.925372306604668271806604899129
