| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 5·11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 6·19-s + 20-s + 21-s + 5·22-s − 4·23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.33169557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.33169557\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.66229774281523, −12.36934405117353, −11.86957510840838, −11.46417906506053, −11.08705968712597, −10.35450103140618, −9.958601242338982, −9.457242764627851, −9.156394557721445, −8.603233967837640, −7.939481412302639, −7.568205253662189, −7.179065770943738, −6.384716817583418, −6.216676244013038, −5.654532223762816, −5.015029633866906, −4.427117034154162, −4.186324707131535, −3.511063448331848, −2.909273391296998, −2.590385621454629, −1.723800702416780, −1.348782400474778, −0.7547983914717458,
0.7547983914717458, 1.348782400474778, 1.723800702416780, 2.590385621454629, 2.909273391296998, 3.511063448331848, 4.186324707131535, 4.427117034154162, 5.015029633866906, 5.654532223762816, 6.216676244013038, 6.384716817583418, 7.179065770943738, 7.568205253662189, 7.939481412302639, 8.603233967837640, 9.156394557721445, 9.457242764627851, 9.958601242338982, 10.35450103140618, 11.08705968712597, 11.46417906506053, 11.86957510840838, 12.36934405117353, 12.66229774281523
