| L(s) = 1 | + 3-s + 9-s − 4·11-s − 2·13-s − 6·19-s − 8·23-s − 5·25-s + 27-s + 8·29-s − 6·31-s − 4·33-s − 37-s − 2·39-s + 2·41-s + 6·43-s − 7·49-s + 2·53-s − 6·57-s + 2·61-s − 8·67-s − 8·69-s − 6·73-s − 5·75-s + 10·79-s + 81-s + 12·83-s + 8·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 1.37·19-s − 1.66·23-s − 25-s + 0.192·27-s + 1.48·29-s − 1.07·31-s − 0.696·33-s − 0.164·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 49-s + 0.274·53-s − 0.794·57-s + 0.256·61-s − 0.977·67-s − 0.963·69-s − 0.702·73-s − 0.577·75-s + 1.12·79-s + 1/9·81-s + 1.31·83-s + 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.790560589892328004891979318941, −8.042425191395369556041130147550, −7.58530583569497081245078307493, −6.50663086574855673577648393459, −5.67681633497955581374856672449, −4.65493893638456750164971260323, −3.86630816709364644791975354443, −2.66910655219050169183424701143, −1.96630478922274497834330535271, 0,
1.96630478922274497834330535271, 2.66910655219050169183424701143, 3.86630816709364644791975354443, 4.65493893638456750164971260323, 5.67681633497955581374856672449, 6.50663086574855673577648393459, 7.58530583569497081245078307493, 8.042425191395369556041130147550, 8.790560589892328004891979318941
