| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 11-s − 2·12-s − 3·13-s + 16-s + 4·17-s − 18-s − 19-s − 22-s − 3·23-s + 2·24-s + 3·26-s + 4·27-s + 5·29-s − 3·31-s − 32-s − 2·33-s − 4·34-s + 36-s + 12·37-s + 38-s + 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.832·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.229·19-s − 0.213·22-s − 0.625·23-s + 0.408·24-s + 0.588·26-s + 0.769·27-s + 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.348·33-s − 0.685·34-s + 1/6·36-s + 1.97·37-s + 0.162·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6304349988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6304349988\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−10.72037843153002036672943444974, −10.04169591442891944937637527504, −9.191320515814848706074123387600, −8.050464216181852944362214228705, −7.20142084928457141587669449399, −6.18203033874227666822982791381, −5.49045025008948680682281531457, −4.26001084862830066585868170883, −2.59263588095010113447027386467, −0.830971576711547037902968231013,
0.830971576711547037902968231013, 2.59263588095010113447027386467, 4.26001084862830066585868170883, 5.49045025008948680682281531457, 6.18203033874227666822982791381, 7.20142084928457141587669449399, 8.050464216181852944362214228705, 9.191320515814848706074123387600, 10.04169591442891944937637527504, 10.72037843153002036672943444974
