| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 2·11-s − 5·13-s + 2·15-s + 2·17-s + 6·19-s − 4·21-s − 4·25-s − 4·27-s − 5·29-s − 4·33-s − 2·35-s − 2·37-s − 10·39-s − 9·41-s − 8·43-s + 45-s + 10·47-s − 3·49-s + 4·51-s + 9·53-s − 2·55-s + 12·57-s − 10·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 0.516·15-s + 0.485·17-s + 1.37·19-s − 0.872·21-s − 4/5·25-s − 0.769·27-s − 0.928·29-s − 0.696·33-s − 0.338·35-s − 0.328·37-s − 1.60·39-s − 1.40·41-s − 1.21·43-s + 0.149·45-s + 1.45·47-s − 3/7·49-s + 0.560·51-s + 1.23·53-s − 0.269·55-s + 1.58·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−7.970149907357089721843662962218, −7.44658662130150213364894603541, −6.80401420910639339110519528843, −5.59270486740101855591008232730, −5.26227374377786082991676771070, −3.97615241556850335032697076190, −3.16306116159616885349758080476, −2.63134163762360014730249483936, −1.72448900237207768978902674922, 0,
1.72448900237207768978902674922, 2.63134163762360014730249483936, 3.16306116159616885349758080476, 3.97615241556850335032697076190, 5.26227374377786082991676771070, 5.59270486740101855591008232730, 6.80401420910639339110519528843, 7.44658662130150213364894603541, 7.970149907357089721843662962218
