| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 6·13-s − 15-s + 2·17-s + 25-s + 27-s + 2·29-s − 4·31-s + 4·33-s + 10·37-s + 6·39-s − 6·41-s − 43-s − 45-s − 7·49-s + 2·51-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s − 6·65-s − 12·67-s + 16·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 0.485·17-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.744·65-s − 1.46·67-s + 1.89·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.890313895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.890313895\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.69684486793198, −14.27692089200213, −13.78559313903565, −13.16433564081912, −12.84331065954755, −12.06441553970147, −11.59843529188309, −11.20056571862358, −10.57413893242340, −9.972945810622699, −9.275475493548638, −8.974566668012512, −8.303971493260803, −7.973917318806164, −7.275920640942862, −6.516985972367761, −6.300394582488312, −5.471847275510535, −4.711420421376437, −3.935684943005344, −3.673895617720552, −3.086136587861677, −2.120487480599975, −1.347745840065911, −0.7707208789212679,
0.7707208789212679, 1.347745840065911, 2.120487480599975, 3.086136587861677, 3.673895617720552, 3.935684943005344, 4.711420421376437, 5.471847275510535, 6.300394582488312, 6.516985972367761, 7.275920640942862, 7.973917318806164, 8.303971493260803, 8.974566668012512, 9.275475493548638, 9.972945810622699, 10.57413893242340, 11.20056571862358, 11.59843529188309, 12.06441553970147, 12.84331065954755, 13.16433564081912, 13.78559313903565, 14.27692089200213, 14.69684486793198
