| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 9-s + 11-s + 2·12-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 18-s − 8·21-s − 22-s − 2·24-s − 2·26-s − 4·27-s − 4·28-s + 6·29-s − 6·31-s − 32-s + 2·33-s − 2·34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.74·21-s − 0.213·22-s − 0.408·24-s − 0.392·26-s − 0.769·27-s − 0.755·28-s + 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7665459650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7665459650\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.12863916515045, −12.68864569229459, −12.14868786504923, −11.65072473256910, −11.15975020479861, −10.48242521789458, −10.02937226262734, −9.753268788345551, −9.136712989246020, −8.980765305945968, −8.276454333615675, −8.099306341489376, −7.334355353974738, −6.960854104238402, −6.364546577015377, −6.055919882973831, −5.401530987945457, −4.611747345419910, −3.837585962029603, −3.433655273062668, −2.984621883314216, −2.660565631306698, −1.713738644734208, −1.334911882762549, −0.2494373115010198,
0.2494373115010198, 1.334911882762549, 1.713738644734208, 2.660565631306698, 2.984621883314216, 3.433655273062668, 3.837585962029603, 4.611747345419910, 5.401530987945457, 6.055919882973831, 6.364546577015377, 6.960854104238402, 7.334355353974738, 8.099306341489376, 8.276454333615675, 8.980765305945968, 9.136712989246020, 9.753268788345551, 10.02937226262734, 10.48242521789458, 11.15975020479861, 11.65072473256910, 12.14868786504923, 12.68864569229459, 13.12863916515045
