| L(s) = 1 | + 3·5-s + 7-s + 5·11-s − 2·13-s − 2·17-s − 6·19-s + 9·23-s + 4·25-s + 10·31-s + 3·35-s − 9·37-s + 9·41-s − 2·43-s − 7·47-s − 6·49-s + 53-s + 15·55-s + 6·61-s − 6·65-s − 2·67-s − 2·71-s + 9·73-s + 5·77-s + 8·79-s + 9·83-s − 6·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 1.50·11-s − 0.554·13-s − 0.485·17-s − 1.37·19-s + 1.87·23-s + 4/5·25-s + 1.79·31-s + 0.507·35-s − 1.47·37-s + 1.40·41-s − 0.304·43-s − 1.02·47-s − 6/7·49-s + 0.137·53-s + 2.02·55-s + 0.768·61-s − 0.744·65-s − 0.244·67-s − 0.237·71-s + 1.05·73-s + 0.569·77-s + 0.900·79-s + 0.987·83-s − 0.650·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.853976121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.853976121\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05828475241714, −12.48321597331672, −12.06391588641139, −11.51148162927621, −11.01397211099255, −10.65220679375000, −10.07485626051040, −9.601978820377046, −9.256029953700376, −8.745984065219709, −8.450436433388605, −7.755531074258621, −6.950674086319359, −6.683368742350894, −6.334441385461602, −5.844454779403151, −5.015897847332486, −4.830759898532574, −4.264162677275872, −3.543471772424616, −2.932728785291338, −2.233040475071809, −1.891715961823610, −1.231324912512979, −0.6281897060225767,
0.6281897060225767, 1.231324912512979, 1.891715961823610, 2.233040475071809, 2.932728785291338, 3.543471772424616, 4.264162677275872, 4.830759898532574, 5.015897847332486, 5.844454779403151, 6.334441385461602, 6.683368742350894, 6.950674086319359, 7.755531074258621, 8.450436433388605, 8.745984065219709, 9.256029953700376, 9.601978820377046, 10.07485626051040, 10.65220679375000, 11.01397211099255, 11.51148162927621, 12.06391588641139, 12.48321597331672, 13.05828475241714
