| L(s) = 1 | + 3·3-s + 2·5-s + 2·7-s + 6·9-s + 3·11-s − 2·13-s + 6·15-s − 6·17-s + 6·21-s + 4·23-s − 25-s + 9·27-s + 6·29-s − 8·31-s + 9·33-s + 4·35-s + 8·37-s − 6·39-s + 3·41-s + 8·43-s + 12·45-s − 10·47-s − 3·49-s − 18·51-s − 2·53-s + 6·55-s − 3·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 0.755·7-s + 2·9-s + 0.904·11-s − 0.554·13-s + 1.54·15-s − 1.45·17-s + 1.30·21-s + 0.834·23-s − 1/5·25-s + 1.73·27-s + 1.11·29-s − 1.43·31-s + 1.56·33-s + 0.676·35-s + 1.31·37-s − 0.960·39-s + 0.468·41-s + 1.21·43-s + 1.78·45-s − 1.45·47-s − 3/7·49-s − 2.52·51-s − 0.274·53-s + 0.809·55-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.092665252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.092665252\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−16.24211688362808, −15.78972575587485, −14.86910636926941, −14.69703915872070, −14.29047719471600, −13.69852141534347, −13.10992103284543, −12.81650178335707, −11.85446136136222, −11.13500163673099, −10.58420375767198, −9.623302383398618, −9.346610202237779, −8.978723389740678, −8.150170870189697, −7.796663356696311, −6.839127744577902, −6.524887387420352, −5.407549114522538, −4.613206378899174, −4.083432665530738, −3.226398903568339, −2.303262825441711, −2.053601849354159, −1.110088295828571,
1.110088295828571, 2.053601849354159, 2.303262825441711, 3.226398903568339, 4.083432665530738, 4.613206378899174, 5.407549114522538, 6.524887387420352, 6.839127744577902, 7.796663356696311, 8.150170870189697, 8.978723389740678, 9.346610202237779, 9.623302383398618, 10.58420375767198, 11.13500163673099, 11.85446136136222, 12.81650178335707, 13.10992103284543, 13.69852141534347, 14.29047719471600, 14.69703915872070, 14.86910636926941, 15.78972575587485, 16.24211688362808
