| L(s) = 1 | + 3-s − 2·9-s − 3·11-s − 4·13-s − 6·17-s + 2·19-s + 6·23-s − 5·25-s − 5·27-s + 6·29-s + 4·31-s − 3·33-s − 37-s − 4·39-s + 9·41-s − 8·43-s − 3·47-s − 6·51-s + 3·53-s + 2·57-s + 12·59-s + 8·61-s + 4·67-s + 6·69-s − 15·71-s − 11·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s − 25-s − 0.962·27-s + 1.11·29-s + 0.718·31-s − 0.522·33-s − 0.164·37-s − 0.640·39-s + 1.40·41-s − 1.21·43-s − 0.437·47-s − 0.840·51-s + 0.412·53-s + 0.264·57-s + 1.56·59-s + 1.02·61-s + 0.488·67-s + 0.722·69-s − 1.78·71-s − 1.28·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9448276458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9448276458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.40099729284733, −13.30810718836472, −12.83151124053123, −12.02463236034565, −11.69252927963540, −11.23892759761145, −10.67843243448490, −10.04458891784477, −9.772577868932005, −9.092851321852672, −8.628832522720217, −8.281597202317017, −7.662207802219348, −7.200413001676259, −6.699130193329357, −6.057336083563729, −5.377539406413000, −4.997937793239295, −4.422547044095398, −3.792493597599646, −2.943397572419529, −2.605167576957341, −2.264174118268016, −1.274907456898013, −0.2817300032892027,
0.2817300032892027, 1.274907456898013, 2.264174118268016, 2.605167576957341, 2.943397572419529, 3.792493597599646, 4.422547044095398, 4.997937793239295, 5.377539406413000, 6.057336083563729, 6.699130193329357, 7.200413001676259, 7.662207802219348, 8.281597202317017, 8.628832522720217, 9.092851321852672, 9.772577868932005, 10.04458891784477, 10.67843243448490, 11.23892759761145, 11.69252927963540, 12.02463236034565, 12.83151124053123, 13.30810718836472, 13.40099729284733
