| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s − 13-s − 14-s + 16-s − 3·17-s − 18-s + 2·19-s − 21-s + 3·23-s + 24-s − 5·25-s + 26-s − 27-s + 28-s + 4·29-s − 2·31-s − 32-s + 3·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s + 0.625·23-s + 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s − 0.359·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−14.64193562505659, −13.92297212530749, −13.32502764446706, −13.00495472409238, −12.22268869880517, −11.76850663636207, −11.49245790331085, −10.79119929527343, −10.55716370222330, −9.794203912661925, −9.453206126292966, −8.881996264733858, −8.267730911797617, −7.747767620065059, −7.242817236338180, −6.755478520191846, −6.045960870146031, −5.713629675342619, −4.860766333260571, −4.498191244915444, −3.721043518628993, −2.904379661819209, −2.279612276610150, −1.542498953660366, −0.8452247636331641, 0,
0.8452247636331641, 1.542498953660366, 2.279612276610150, 2.904379661819209, 3.721043518628993, 4.498191244915444, 4.860766333260571, 5.713629675342619, 6.045960870146031, 6.755478520191846, 7.242817236338180, 7.747767620065059, 8.267730911797617, 8.881996264733858, 9.453206126292966, 9.794203912661925, 10.55716370222330, 10.79119929527343, 11.49245790331085, 11.76850663636207, 12.22268869880517, 13.00495472409238, 13.32502764446706, 13.92297212530749, 14.64193562505659
