| L(s) = 1 | − 3·5-s − 3·11-s + 13-s + 7·17-s − 19-s + 23-s + 4·25-s + 2·29-s + 9·31-s − 3·37-s + 10·41-s − 4·43-s − 3·47-s + 53-s + 9·55-s + 11·59-s + 61-s − 3·65-s + 7·67-s + 8·71-s + 7·73-s + 11·79-s + 4·83-s − 21·85-s + 89-s + 3·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.904·11-s + 0.277·13-s + 1.69·17-s − 0.229·19-s + 0.208·23-s + 4/5·25-s + 0.371·29-s + 1.61·31-s − 0.493·37-s + 1.56·41-s − 0.609·43-s − 0.437·47-s + 0.137·53-s + 1.21·55-s + 1.43·59-s + 0.128·61-s − 0.372·65-s + 0.855·67-s + 0.949·71-s + 0.819·73-s + 1.23·79-s + 0.439·83-s − 2.27·85-s + 0.105·89-s + 0.307·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138197251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138197251\) |
| \(L(\frac{3}{2})\) |
|
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.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 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
−13.94529852298001, −13.28704582440401, −12.74290177470282, −12.33982155537118, −11.90572560013058, −11.43442800721845, −10.95864758121352, −10.38567189655935, −9.954706420298432, −9.467002127337950, −8.519366612202992, −8.321265890178877, −7.808449541905720, −7.457085089490773, −6.819609993450875, −6.190147443038449, −5.580777510024827, −4.937775767471102, −4.570936712694451, −3.677955428446089, −3.481121845625436, −2.757090919176756, −2.092653800842763, −0.9256587508494869, −0.6126877421932328,
0.6126877421932328, 0.9256587508494869, 2.092653800842763, 2.757090919176756, 3.481121845625436, 3.677955428446089, 4.570936712694451, 4.937775767471102, 5.580777510024827, 6.190147443038449, 6.819609993450875, 7.457085089490773, 7.808449541905720, 8.321265890178877, 8.519366612202992, 9.467002127337950, 9.954706420298432, 10.38567189655935, 10.95864758121352, 11.43442800721845, 11.90572560013058, 12.33982155537118, 12.74290177470282, 13.28704582440401, 13.94529852298001
