| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 5·11-s + 12-s + 6·13-s + 14-s + 16-s − 17-s + 18-s + 7·19-s + 21-s − 5·22-s + 6·23-s + 24-s + 6·26-s + 27-s + 28-s − 5·29-s + 7·31-s + 32-s − 5·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.60·19-s + 0.218·21-s − 1.06·22-s + 1.25·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.870·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.378902713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.378902713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 17 T + p T^{2} \) | 1.83.ar |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.65504488450462, −15.27419376297788, −14.78811203332485, −13.99881637971742, −13.47665478937308, −13.28894974059245, −12.83556823354542, −11.76137045189631, −11.56348123599494, −10.80274758275378, −10.34293925177767, −9.725424244240287, −8.795891119475131, −8.467062044016955, −7.729523074245773, −7.297144002333892, −6.536834112492215, −5.798015351480800, −5.114656466723232, −4.786867451449186, −3.671325266180445, −3.299843002069945, −2.610068342310967, −1.717738684521647, −0.8841941520379270,
0.8841941520379270, 1.717738684521647, 2.610068342310967, 3.299843002069945, 3.671325266180445, 4.786867451449186, 5.114656466723232, 5.798015351480800, 6.536834112492215, 7.297144002333892, 7.729523074245773, 8.467062044016955, 8.795891119475131, 9.725424244240287, 10.34293925177767, 10.80274758275378, 11.56348123599494, 11.76137045189631, 12.83556823354542, 13.28894974059245, 13.47665478937308, 13.99881637971742, 14.78811203332485, 15.27419376297788, 15.65504488450462
