| L(s) = 1 | + 3·5-s + 11-s + 13-s + 5·17-s − 3·19-s + 9·23-s + 4·25-s − 6·29-s − 7·31-s − 7·37-s − 6·41-s − 12·43-s − 7·47-s − 9·53-s + 3·55-s − 3·59-s + 5·61-s + 3·65-s + 3·67-s + 8·71-s − 13·73-s + 15·79-s − 4·83-s + 15·85-s + 13·89-s − 9·95-s + 14·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.301·11-s + 0.277·13-s + 1.21·17-s − 0.688·19-s + 1.87·23-s + 4/5·25-s − 1.11·29-s − 1.25·31-s − 1.15·37-s − 0.937·41-s − 1.82·43-s − 1.02·47-s − 1.23·53-s + 0.404·55-s − 0.390·59-s + 0.640·61-s + 0.372·65-s + 0.366·67-s + 0.949·71-s − 1.52·73-s + 1.68·79-s − 0.439·83-s + 1.62·85-s + 1.37·89-s − 0.923·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.223679828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.223679828\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.63349601897186, −12.13281816452430, −11.51128929474076, −11.11294528200191, −10.68319706144528, −10.17462237259257, −9.778869815280174, −9.416793939849349, −8.824569151527733, −8.659065601192446, −7.904956896721881, −7.421202967395716, −6.810751457435592, −6.539127641712860, −5.991423671916637, −5.456791552274023, −5.050495154803086, −4.805545233513491, −3.706057100082151, −3.447470613161656, −2.987179448861686, −2.097270966325994, −1.702878455470763, −1.351070804432524, −0.4407533969535785,
0.4407533969535785, 1.351070804432524, 1.702878455470763, 2.097270966325994, 2.987179448861686, 3.447470613161656, 3.706057100082151, 4.805545233513491, 5.050495154803086, 5.456791552274023, 5.991423671916637, 6.539127641712860, 6.810751457435592, 7.421202967395716, 7.904956896721881, 8.659065601192446, 8.824569151527733, 9.416793939849349, 9.778869815280174, 10.17462237259257, 10.68319706144528, 11.11294528200191, 11.51128929474076, 12.13281816452430, 12.63349601897186
