| L(s) = 1 | + 3-s − 7-s − 2·9-s + 11-s − 2·13-s + 3·17-s + 19-s − 21-s + 6·23-s − 5·27-s − 9·29-s − 5·31-s + 33-s − 5·37-s − 2·39-s − 6·41-s + 8·43-s + 6·47-s − 6·49-s + 3·51-s − 9·53-s + 57-s − 6·59-s + 5·61-s + 2·63-s + 8·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.554·13-s + 0.727·17-s + 0.229·19-s − 0.218·21-s + 1.25·23-s − 0.962·27-s − 1.67·29-s − 0.898·31-s + 0.174·33-s − 0.821·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s − 6/7·49-s + 0.420·51-s − 1.23·53-s + 0.132·57-s − 0.781·59-s + 0.640·61-s + 0.251·63-s + 0.977·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.006482860188793392060803299705, −7.32205453480294128749479591732, −6.69469424367850562548093274341, −5.60201323400911262634557654835, −5.23949695535708623688309713356, −3.97503625336556497409288224469, −3.31395620758984011122159022720, −2.59161138645642019963353373167, −1.50749609303058720925151531805, 0,
1.50749609303058720925151531805, 2.59161138645642019963353373167, 3.31395620758984011122159022720, 3.97503625336556497409288224469, 5.23949695535708623688309713356, 5.60201323400911262634557654835, 6.69469424367850562548093274341, 7.32205453480294128749479591732, 8.006482860188793392060803299705
