| L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s − 2·13-s − 17-s − 2·19-s − 21-s + 6·23-s − 27-s − 2·29-s + 10·31-s − 4·33-s + 8·37-s + 2·39-s + 8·43-s − 4·47-s + 49-s + 51-s + 6·53-s + 2·57-s − 4·59-s + 2·61-s + 63-s + 8·67-s − 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.458·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.696·33-s + 1.31·37-s + 0.320·39-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.264·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s + 0.977·67-s − 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398303243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398303243\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.01016524341719, −14.44796929848496, −13.90305110975056, −13.35748504834964, −12.65184984735824, −12.35134405025080, −11.63939446835283, −11.30548295410992, −10.85415192520177, −10.14114932664082, −9.560874880213539, −9.159363359559158, −8.447971834903163, −7.920382260311060, −7.154334336633639, −6.721999339880969, −6.198663837182003, −5.558605409181536, −4.808104185872950, −4.402144648791777, −3.799984206504740, −2.852694998950903, −2.198799574849160, −1.239030023514372, −0.6773982913150067,
0.6773982913150067, 1.239030023514372, 2.198799574849160, 2.852694998950903, 3.799984206504740, 4.402144648791777, 4.808104185872950, 5.558605409181536, 6.198663837182003, 6.721999339880969, 7.154334336633639, 7.920382260311060, 8.447971834903163, 9.159363359559158, 9.560874880213539, 10.14114932664082, 10.85415192520177, 11.30548295410992, 11.63939446835283, 12.35134405025080, 12.65184984735824, 13.35748504834964, 13.90305110975056, 14.44796929848496, 15.01016524341719
