| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 11-s + 2·14-s + 16-s − 2·17-s − 2·19-s + 22-s + 4·23-s − 2·28-s + 29-s − 32-s + 2·34-s + 2·37-s + 2·38-s + 6·41-s + 6·43-s − 44-s − 4·46-s + 10·47-s − 3·49-s − 6·53-s + 2·56-s − 58-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.301·11-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s + 0.213·22-s + 0.834·23-s − 0.377·28-s + 0.185·29-s − 0.176·32-s + 0.342·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s + 0.914·43-s − 0.150·44-s − 0.589·46-s + 1.45·47-s − 3/7·49-s − 0.824·53-s + 0.267·56-s − 0.131·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637822972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637822972\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 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.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.12803498735274, −12.91221260766973, −12.51085540081045, −11.93502405626545, −11.27063511480346, −10.93782823481314, −10.55343430529897, −9.889166454280401, −9.557258555432025, −9.036305696570866, −8.646077137025164, −8.054622023706311, −7.531598381658208, −7.048100972409525, −6.524464361296165, −6.122998132728803, −5.523807952616080, −4.894961886485273, −4.259979795912952, −3.632602327268795, −3.045903833068839, −2.395461336715131, −2.020836979578647, −0.9082749751532866, −0.5367461898556492,
0.5367461898556492, 0.9082749751532866, 2.020836979578647, 2.395461336715131, 3.045903833068839, 3.632602327268795, 4.259979795912952, 4.894961886485273, 5.523807952616080, 6.122998132728803, 6.524464361296165, 7.048100972409525, 7.531598381658208, 8.054622023706311, 8.646077137025164, 9.036305696570866, 9.557258555432025, 9.889166454280401, 10.55343430529897, 10.93782823481314, 11.27063511480346, 11.93502405626545, 12.51085540081045, 12.91221260766973, 13.12803498735274
