| L(s) = 1 | − 3-s − 3·5-s − 2·9-s + 3·11-s + 3·15-s + 6·17-s + 19-s + 4·25-s + 5·27-s − 3·29-s − 5·31-s − 3·33-s − 10·37-s − 9·41-s − 5·43-s + 6·45-s − 3·47-s − 6·51-s + 9·53-s − 9·55-s − 57-s − 12·59-s + 5·61-s − 5·67-s − 15·71-s − 73-s − 4·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.904·11-s + 0.774·15-s + 1.45·17-s + 0.229·19-s + 4/5·25-s + 0.962·27-s − 0.557·29-s − 0.898·31-s − 0.522·33-s − 1.64·37-s − 1.40·41-s − 0.762·43-s + 0.894·45-s − 0.437·47-s − 0.840·51-s + 1.23·53-s − 1.21·55-s − 0.132·57-s − 1.56·59-s + 0.640·61-s − 0.610·67-s − 1.78·71-s − 0.117·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.83542931981088, −13.66687265511371, −12.73791156298758, −12.21058915059037, −12.05995312024811, −11.55950448806458, −11.30765653528386, −10.62471888904773, −10.23503446362124, −9.602996535168238, −8.975668996719465, −8.520058501704797, −8.130962491321734, −7.489878317387310, −7.075482580570052, −6.628127299384527, −5.887554500115270, −5.399814851660463, −5.046632588456501, −4.216882893487097, −3.746872884438194, −3.308186967357195, −2.808342734242283, −1.609680021078974, −1.225918980480055, 0, 0,
1.225918980480055, 1.609680021078974, 2.808342734242283, 3.308186967357195, 3.746872884438194, 4.216882893487097, 5.046632588456501, 5.399814851660463, 5.887554500115270, 6.628127299384527, 7.075482580570052, 7.489878317387310, 8.130962491321734, 8.520058501704797, 8.975668996719465, 9.602996535168238, 10.23503446362124, 10.62471888904773, 11.30765653528386, 11.55950448806458, 12.05995312024811, 12.21058915059037, 12.73791156298758, 13.66687265511371, 13.83542931981088
