| L(s) = 1 | − 3-s + 5-s − 2·9-s + 4·11-s − 13-s − 15-s + 3·17-s + 5·23-s − 4·25-s + 5·27-s + 7·29-s − 4·31-s − 4·33-s + 10·37-s + 39-s + 5·41-s + 5·43-s − 2·45-s − 7·47-s − 7·49-s − 3·51-s + 11·53-s + 4·55-s + 3·59-s − 11·61-s − 65-s − 3·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s + 1.04·23-s − 4/5·25-s + 0.962·27-s + 1.29·29-s − 0.718·31-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 0.780·41-s + 0.762·43-s − 0.298·45-s − 1.02·47-s − 49-s − 0.420·51-s + 1.51·53-s + 0.539·55-s + 0.390·59-s − 1.40·61-s − 0.124·65-s − 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.158788461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.158788461\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.40724024509656, −14.80177119808262, −14.38175471456773, −14.00871036210535, −13.29372746570438, −12.69514908208697, −12.12604742295437, −11.64108299370893, −11.23096420694122, −10.58991632217794, −9.982657784867422, −9.295426404617514, −9.057283462922782, −8.191960104144640, −7.632646534905688, −6.868226462943132, −6.231987865230720, −5.937223502618164, −5.173788115316847, −4.610200921438778, −3.812570257583015, −3.050785870512505, −2.362741210672631, −1.338586666823300, −0.6696474370087774,
0.6696474370087774, 1.338586666823300, 2.362741210672631, 3.050785870512505, 3.812570257583015, 4.610200921438778, 5.173788115316847, 5.937223502618164, 6.231987865230720, 6.868226462943132, 7.632646534905688, 8.191960104144640, 9.057283462922782, 9.295426404617514, 9.982657784867422, 10.58991632217794, 11.23096420694122, 11.64108299370893, 12.12604742295437, 12.69514908208697, 13.29372746570438, 14.00871036210535, 14.38175471456773, 14.80177119808262, 15.40724024509656
