| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 3·13-s + 15-s − 7·17-s − 21-s + 5·23-s + 25-s + 5·27-s + 5·29-s + 10·31-s − 35-s − 2·37-s − 3·39-s − 2·41-s − 6·43-s + 2·45-s − 6·49-s + 7·51-s − 9·53-s − 7·59-s − 4·61-s − 2·63-s − 3·65-s + 7·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.832·13-s + 0.258·15-s − 1.69·17-s − 0.218·21-s + 1.04·23-s + 1/5·25-s + 0.962·27-s + 0.928·29-s + 1.79·31-s − 0.169·35-s − 0.328·37-s − 0.480·39-s − 0.312·41-s − 0.914·43-s + 0.298·45-s − 6/7·49-s + 0.980·51-s − 1.23·53-s − 0.911·59-s − 0.512·61-s − 0.251·63-s − 0.372·65-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−15.54244768443002, −15.06925644077513, −14.30167511706418, −13.92198788048826, −13.22710491431836, −12.90567030380054, −11.98631799694856, −11.68196385211296, −11.23880549216133, −10.70978615021663, −10.33544387655528, −9.364792258889251, −8.806659843691288, −8.370338162483946, −7.937286452002815, −6.978933162590564, −6.434903321798213, −6.201424552902223, −5.164821157204875, −4.749226287404171, −4.260814980005370, −3.223315568545001, −2.809033372727354, −1.774328404750973, −0.9059875359341912, 0,
0.9059875359341912, 1.774328404750973, 2.809033372727354, 3.223315568545001, 4.260814980005370, 4.749226287404171, 5.164821157204875, 6.201424552902223, 6.434903321798213, 6.978933162590564, 7.937286452002815, 8.370338162483946, 8.806659843691288, 9.364792258889251, 10.33544387655528, 10.70978615021663, 11.23880549216133, 11.68196385211296, 11.98631799694856, 12.90567030380054, 13.22710491431836, 13.92198788048826, 14.30167511706418, 15.06925644077513, 15.54244768443002
