| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 3·9-s − 10-s + 6·11-s − 2·13-s + 16-s + 17-s + 3·18-s − 8·19-s + 20-s − 6·22-s − 4·23-s + 25-s + 2·26-s + 10·29-s − 8·31-s − 32-s − 34-s − 3·36-s − 8·37-s + 8·38-s − 40-s − 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 9-s − 0.316·10-s + 1.80·11-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.83·19-s + 0.223·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s − 1/2·36-s − 1.31·37-s + 1.29·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291882733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291882733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.990279878657629552975281664036, −6.97118270012003083899240157204, −6.50979408493889219597540636146, −5.98258515296966971538137664572, −5.13912677655092839398803122604, −4.15029940975193900477955130754, −3.45061960880483354722621453028, −2.36557022294362102988764227886, −1.79960277469354253606180983882, −0.61076295949644698555682830847,
0.61076295949644698555682830847, 1.79960277469354253606180983882, 2.36557022294362102988764227886, 3.45061960880483354722621453028, 4.15029940975193900477955130754, 5.13912677655092839398803122604, 5.98258515296966971538137664572, 6.50979408493889219597540636146, 6.97118270012003083899240157204, 7.990279878657629552975281664036
