| L(s) = 1 | − 5-s + 2·13-s + 17-s + 4·19-s + 25-s + 10·29-s + 8·31-s + 6·37-s + 2·41-s − 8·43-s − 7·49-s + 2·53-s + 12·59-s − 14·61-s − 2·65-s − 8·67-s + 4·71-s + 6·73-s + 16·79-s − 4·83-s − 85-s + 6·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s + 0.242·17-s + 0.917·19-s + 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 1.21·43-s − 49-s + 0.274·53-s + 1.56·59-s − 1.79·61-s − 0.248·65-s − 0.977·67-s + 0.474·71-s + 0.702·73-s + 1.80·79-s − 0.439·83-s − 0.108·85-s + 0.635·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.709780410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709780410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.53450384026312, −13.95918855324840, −13.60029262517112, −13.06969886723741, −12.39495893780880, −11.86672186105417, −11.64405739317522, −10.93400728628511, −10.41448080124309, −9.854881783917473, −9.432318385179104, −8.611379770827930, −8.254404790839037, −7.777275609848429, −7.133204670125455, −6.425076741956013, −6.156859698507174, −5.259121337465890, −4.719789777462673, −4.238492567662793, −3.316245933223446, −3.050536237739556, −2.170730567418665, −1.175172167392695, −0.6730032063198411,
0.6730032063198411, 1.175172167392695, 2.170730567418665, 3.050536237739556, 3.316245933223446, 4.238492567662793, 4.719789777462673, 5.259121337465890, 6.156859698507174, 6.425076741956013, 7.133204670125455, 7.777275609848429, 8.254404790839037, 8.611379770827930, 9.432318385179104, 9.854881783917473, 10.41448080124309, 10.93400728628511, 11.64405739317522, 11.86672186105417, 12.39495893780880, 13.06969886723741, 13.60029262517112, 13.95918855324840, 14.53450384026312
