| L(s) = 1 | − 3·3-s − 3·5-s + 6·9-s + 4·11-s + 9·15-s − 5·17-s − 6·19-s − 6·23-s + 4·25-s − 9·27-s − 4·29-s − 12·33-s + 3·37-s + 12·41-s − 3·43-s − 18·45-s − 7·47-s + 15·51-s − 2·53-s − 12·55-s + 18·57-s + 2·59-s + 12·61-s + 4·67-s + 18·69-s − 11·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 2·9-s + 1.20·11-s + 2.32·15-s − 1.21·17-s − 1.37·19-s − 1.25·23-s + 4/5·25-s − 1.73·27-s − 0.742·29-s − 2.08·33-s + 0.493·37-s + 1.87·41-s − 0.457·43-s − 2.68·45-s − 1.02·47-s + 2.10·51-s − 0.274·53-s − 1.61·55-s + 2.38·57-s + 0.260·59-s + 1.53·61-s + 0.488·67-s + 2.16·69-s − 1.30·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.81681895842949, −12.53050183887396, −11.86293294271180, −11.62217844959962, −11.26885482661112, −11.03888658269220, −10.40174584882108, −10.00273556375758, −9.312209001740837, −8.910238989066779, −8.226779089908982, −7.881203753681238, −7.212445799527558, −6.759494259786863, −6.449303102639595, −5.980025949998583, −5.504537668270406, −4.638877678397305, −4.452431856416606, −3.887289267641115, −3.767184167430178, −2.587425528043060, −1.914097911813065, −1.254687214151496, −0.4678913926091426, 0,
0.4678913926091426, 1.254687214151496, 1.914097911813065, 2.587425528043060, 3.767184167430178, 3.887289267641115, 4.452431856416606, 4.638877678397305, 5.504537668270406, 5.980025949998583, 6.449303102639595, 6.759494259786863, 7.212445799527558, 7.881203753681238, 8.226779089908982, 8.910238989066779, 9.312209001740837, 10.00273556375758, 10.40174584882108, 11.03888658269220, 11.26885482661112, 11.62217844959962, 11.86293294271180, 12.53050183887396, 12.81681895842949
