| L(s) = 1 | + 3-s + 9-s − 6·13-s − 2·17-s − 4·19-s + 4·23-s + 27-s − 6·29-s + 6·37-s − 6·39-s + 2·41-s + 4·43-s − 8·47-s − 2·51-s − 2·53-s − 4·57-s + 12·59-s + 6·61-s + 4·67-s + 4·69-s + 12·71-s + 10·73-s + 8·79-s + 81-s − 12·83-s − 6·87-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.986·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s − 0.280·51-s − 0.274·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764656689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764656689\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.92936183482511, −12.58126123937350, −12.09020564208676, −11.46487268224074, −10.95379890156464, −10.75828122735755, −9.785257793539634, −9.691531084167416, −9.350836453556093, −8.583681481269131, −8.263298844154547, −7.746405121877473, −7.210648423223445, −6.798214866500853, −6.428637007418048, −5.476335054791451, −5.278843366564529, −4.530497726551482, −4.183082474170732, −3.585625164368980, −2.871163642334820, −2.343850784573075, −2.085762372108251, −1.175748685157772, −0.3538777843105362,
0.3538777843105362, 1.175748685157772, 2.085762372108251, 2.343850784573075, 2.871163642334820, 3.585625164368980, 4.183082474170732, 4.530497726551482, 5.278843366564529, 5.476335054791451, 6.428637007418048, 6.798214866500853, 7.210648423223445, 7.746405121877473, 8.263298844154547, 8.583681481269131, 9.350836453556093, 9.691531084167416, 9.785257793539634, 10.75828122735755, 10.95379890156464, 11.46487268224074, 12.09020564208676, 12.58126123937350, 12.92936183482511
