| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s + 4·13-s − 16-s − 4·17-s + 18-s + 8·19-s − 22-s − 8·23-s + 3·24-s − 5·25-s + 4·26-s − 27-s + 2·29-s + 4·31-s + 5·32-s + 33-s − 4·34-s − 36-s + 10·37-s + 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.174·33-s − 0.685·34-s − 1/6·36-s + 1.64·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533844148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533844148\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−9.552454693622765888389394766714, −8.553909793382580544364980366756, −7.85134258597300221575314260926, −6.71914881298175970847803262695, −5.83570011756075485976483067875, −5.44248510829291586729906519635, −4.27742942271747976447206821512, −3.79403693919376686631240969854, −2.49490428968644150893949998293, −0.811628513945363488373168988408,
0.811628513945363488373168988408, 2.49490428968644150893949998293, 3.79403693919376686631240969854, 4.27742942271747976447206821512, 5.44248510829291586729906519635, 5.83570011756075485976483067875, 6.71914881298175970847803262695, 7.85134258597300221575314260926, 8.553909793382580544364980366756, 9.552454693622765888389394766714
