| L(s) = 1 | − 2·7-s − 2·13-s + 3·17-s + 5·19-s − 3·23-s − 6·29-s + 5·31-s − 2·37-s + 12·41-s − 8·43-s + 12·47-s − 3·49-s + 3·53-s + 6·59-s − 7·61-s − 2·67-s + 12·71-s + 16·73-s − 79-s + 15·83-s − 12·89-s + 4·91-s + 16·97-s + 12·101-s + 4·103-s + 12·107-s − 7·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.554·13-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 1.11·29-s + 0.898·31-s − 0.328·37-s + 1.87·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s + 0.412·53-s + 0.781·59-s − 0.896·61-s − 0.244·67-s + 1.42·71-s + 1.87·73-s − 0.112·79-s + 1.64·83-s − 1.27·89-s + 0.419·91-s + 1.62·97-s + 1.19·101-s + 0.394·103-s + 1.16·107-s − 0.670·109-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
| \(L(1)\) |
\(\approx\) |
\(1.542881379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542881379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−8.991575858757247606393309220706, −7.897305160082441963348330982303, −7.43651569639532072716364031587, −6.53161336511635401696768522395, −5.75302624148388270241120747668, −5.03755304482550874917001344255, −3.92770459206472377204437706043, −3.18676671219872013299517076275, −2.19940182521032133994309509113, −0.76859025356093006158943339130,
0.76859025356093006158943339130, 2.19940182521032133994309509113, 3.18676671219872013299517076275, 3.92770459206472377204437706043, 5.03755304482550874917001344255, 5.75302624148388270241120747668, 6.53161336511635401696768522395, 7.43651569639532072716364031587, 7.897305160082441963348330982303, 8.991575858757247606393309220706
