| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 5·11-s + 6·13-s + 2·15-s + 3·17-s − 7·19-s + 4·21-s + 2·23-s − 25-s − 27-s + 6·29-s − 2·31-s − 5·33-s + 8·35-s − 4·37-s − 6·39-s + 11·41-s − 2·45-s + 10·47-s + 9·49-s − 3·51-s − 6·53-s − 10·55-s + 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.50·11-s + 1.66·13-s + 0.516·15-s + 0.727·17-s − 1.60·19-s + 0.872·21-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.870·33-s + 1.35·35-s − 0.657·37-s − 0.960·39-s + 1.71·41-s − 0.298·45-s + 1.45·47-s + 9/7·49-s − 0.420·51-s − 0.824·53-s − 1.34·55-s + 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.771597485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.771597485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−12.34606253989720, −12.26612969402279, −11.68356860458177, −11.05779701268169, −10.89912085082214, −10.36632538226872, −9.841551552315926, −9.292083146643460, −8.840804456845605, −8.635797658996536, −7.903090648294243, −7.381438527763880, −6.868542444000760, −6.356423484919302, −6.118433336075120, −5.888157990095292, −4.923797243571471, −4.297002102886745, −3.912399318875655, −3.598078013934960, −3.156859559094732, −2.346971970528212, −1.536584673977847, −0.8793268919125991, −0.4758040391270677,
0.4758040391270677, 0.8793268919125991, 1.536584673977847, 2.346971970528212, 3.156859559094732, 3.598078013934960, 3.912399318875655, 4.297002102886745, 4.923797243571471, 5.888157990095292, 6.118433336075120, 6.356423484919302, 6.868542444000760, 7.381438527763880, 7.903090648294243, 8.635797658996536, 8.840804456845605, 9.292083146643460, 9.841551552315926, 10.36632538226872, 10.89912085082214, 11.05779701268169, 11.68356860458177, 12.26612969402279, 12.34606253989720
