| L(s) = 1 | − 2·5-s − 2·7-s + 11-s + 2·13-s + 2·19-s − 25-s − 8·29-s − 4·31-s + 4·35-s + 6·37-s + 4·41-s + 6·43-s − 8·47-s − 3·49-s − 6·53-s − 2·55-s + 4·59-s + 6·61-s − 4·65-s + 4·67-s − 8·71-s − 10·73-s − 2·77-s + 14·79-s + 8·83-s − 2·89-s − 4·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 1/5·25-s − 1.48·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s + 0.624·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.227·77-s + 1.57·79-s + 0.878·83-s − 0.211·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159947346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159947346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.81375506924883329814512959302, −7.52101383793793774305346433608, −6.60013265187412192040134496781, −6.00715105147020958206847240899, −5.20986180312562036878248115139, −4.16665702556226649237695124076, −3.68724547339914809803628532317, −2.97641658275604673309547181031, −1.79406685802117218926139785798, −0.55418663748464369189567740538,
0.55418663748464369189567740538, 1.79406685802117218926139785798, 2.97641658275604673309547181031, 3.68724547339914809803628532317, 4.16665702556226649237695124076, 5.20986180312562036878248115139, 6.00715105147020958206847240899, 6.60013265187412192040134496781, 7.52101383793793774305346433608, 7.81375506924883329814512959302
