| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s + 2·19-s − 23-s − 27-s − 8·29-s + 10·31-s + 4·33-s + 3·37-s + 39-s + 4·41-s + 10·43-s − 3·47-s − 7·49-s − 6·53-s − 2·57-s + 7·59-s − 61-s − 2·67-s + 69-s + 71-s + 11·73-s + 2·79-s + 81-s − 11·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.458·19-s − 0.208·23-s − 0.192·27-s − 1.48·29-s + 1.79·31-s + 0.696·33-s + 0.493·37-s + 0.160·39-s + 0.624·41-s + 1.52·43-s − 0.437·47-s − 49-s − 0.824·53-s − 0.264·57-s + 0.911·59-s − 0.128·61-s − 0.244·67-s + 0.120·69-s + 0.118·71-s + 1.28·73-s + 0.225·79-s + 1/9·81-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.11183090725132, −13.59133822491208, −13.09447131371704, −12.65094042493740, −12.27507784097430, −11.60469492065033, −11.10543527985762, −10.87010473443123, −10.10398957605078, −9.750334888985331, −9.359810716991520, −8.507307409134504, −8.010155037329745, −7.578334756591408, −7.108048613112878, −6.362374524278411, −5.925054176336423, −5.353402396112525, −4.902475777630832, −4.314762540533048, −3.686870155224600, −2.831975696884181, −2.457289895773834, −1.585068030472029, −0.7710093548772571, 0,
0.7710093548772571, 1.585068030472029, 2.457289895773834, 2.831975696884181, 3.686870155224600, 4.314762540533048, 4.902475777630832, 5.353402396112525, 5.925054176336423, 6.362374524278411, 7.108048613112878, 7.578334756591408, 8.010155037329745, 8.507307409134504, 9.359810716991520, 9.750334888985331, 10.10398957605078, 10.87010473443123, 11.10543527985762, 11.60469492065033, 12.27507784097430, 12.65094042493740, 13.09447131371704, 13.59133822491208, 14.11183090725132
