| L(s) = 1 | − 5·7-s + 7·13-s − 19-s − 4·31-s + 37-s − 8·43-s + 18·49-s − 13·61-s − 11·67-s − 17·73-s − 13·79-s − 35·91-s − 5·97-s + 7·103-s + 2·109-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 1.94·13-s − 0.229·19-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s − 1.66·61-s − 1.34·67-s − 1.98·73-s − 1.46·79-s − 3.66·91-s − 0.507·97-s + 0.689·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−8.759514901282035890818105943566, −7.64362636112665563331595399405, −6.74728657923342957215579477924, −6.19215897625319522417122883742, −5.65654369161366776038771212095, −4.27187623582428822445319431066, −3.50441900239116663157549394317, −2.91544699469635223918630301779, −1.43883911512169505886541205101, 0,
1.43883911512169505886541205101, 2.91544699469635223918630301779, 3.50441900239116663157549394317, 4.27187623582428822445319431066, 5.65654369161366776038771212095, 6.19215897625319522417122883742, 6.74728657923342957215579477924, 7.64362636112665563331595399405, 8.759514901282035890818105943566
