| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 11-s − 2·13-s + 2·14-s + 16-s − 17-s − 19-s + 22-s − 6·23-s − 2·26-s + 2·28-s + 6·29-s − 2·31-s + 32-s − 34-s + 10·37-s − 38-s − 7·41-s − 8·43-s + 44-s − 6·46-s + 8·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.229·19-s + 0.213·22-s − 1.25·23-s − 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.171·34-s + 1.64·37-s − 0.162·38-s − 1.09·41-s − 1.21·43-s + 0.150·44-s − 0.884·46-s + 1.16·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.12292844213393, −14.60203313394821, −14.09041145916206, −13.81534007643169, −13.08187776008832, −12.57639746752706, −11.99902132707983, −11.66243813595973, −11.07068610533305, −10.55553448266979, −9.889818721149972, −9.459735846669815, −8.604562316939191, −8.023393428842065, −7.749509187445517, −6.765230703195447, −6.533278085319314, −5.724924964861925, −5.183469026922814, −4.477954714438374, −4.201799112675585, −3.313780810696210, −2.608932136856015, −1.925300688236193, −1.248369813487436, 0,
1.248369813487436, 1.925300688236193, 2.608932136856015, 3.313780810696210, 4.201799112675585, 4.477954714438374, 5.183469026922814, 5.724924964861925, 6.533278085319314, 6.765230703195447, 7.749509187445517, 8.023393428842065, 8.604562316939191, 9.459735846669815, 9.889818721149972, 10.55553448266979, 11.07068610533305, 11.66243813595973, 11.99902132707983, 12.57639746752706, 13.08187776008832, 13.81534007643169, 14.09041145916206, 14.60203313394821, 15.12292844213393
