| L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s + 2·11-s + 5·13-s + 2·14-s + 16-s − 17-s + 20-s − 2·22-s + 4·23-s − 4·25-s − 5·26-s − 2·28-s − 3·29-s − 2·31-s − 32-s + 34-s − 2·35-s − 7·37-s − 40-s + 6·43-s + 2·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 1.38·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.223·20-s − 0.426·22-s + 0.834·23-s − 4/5·25-s − 0.980·26-s − 0.377·28-s − 0.557·29-s − 0.359·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s − 1.15·37-s − 0.158·40-s + 0.914·43-s + 0.301·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−12.96277530920448, −12.68097678916582, −11.86626875572791, −11.64592472311590, −11.01462344182687, −10.71549818927571, −10.21242167112237, −9.674005554597529, −9.204395404139443, −9.043973795064812, −8.414301057726651, −8.015825449707057, −7.336804794413832, −6.772750352882069, −6.567987241395688, −5.988099742756433, −5.545076992347046, −5.049906180940536, −4.085275350546295, −3.701836391593503, −3.330267140788164, −2.510483221554784, −2.019488706706194, −1.346199255392875, −0.8332668126755916, 0,
0.8332668126755916, 1.346199255392875, 2.019488706706194, 2.510483221554784, 3.330267140788164, 3.701836391593503, 4.085275350546295, 5.049906180940536, 5.545076992347046, 5.988099742756433, 6.567987241395688, 6.772750352882069, 7.336804794413832, 8.015825449707057, 8.414301057726651, 9.043973795064812, 9.204395404139443, 9.674005554597529, 10.21242167112237, 10.71549818927571, 11.01462344182687, 11.64592472311590, 11.86626875572791, 12.68097678916582, 12.96277530920448
