| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 25-s − 27-s − 2·29-s + 2·37-s + 2·39-s − 2·41-s − 12·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s − 6·53-s + 4·57-s + 12·59-s + 6·61-s + 2·65-s − 4·67-s + 6·73-s − 75-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.328·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.768·61-s + 0.248·65-s − 0.488·67-s + 0.702·73-s − 0.115·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6954037423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6954037423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.38558974447904, −13.12948057558481, −12.58665309417458, −12.08862746568252, −11.70009518733754, −11.14232578857459, −10.83840613720542, −10.16024356937611, −9.851917656594586, −9.162247709820616, −8.677291651152407, −8.109410624294280, −7.668770084386360, −7.009290245164351, −6.604720135327725, −6.163193222186388, −5.443426485110183, −4.851013243159046, −4.588171474854057, −3.744778806346024, −3.432250078691781, −2.403941883913257, −2.040079124264690, −1.115995767525526, −0.2872366322631839,
0.2872366322631839, 1.115995767525526, 2.040079124264690, 2.403941883913257, 3.432250078691781, 3.744778806346024, 4.588171474854057, 4.851013243159046, 5.443426485110183, 6.163193222186388, 6.604720135327725, 7.009290245164351, 7.668770084386360, 8.109410624294280, 8.677291651152407, 9.162247709820616, 9.851917656594586, 10.16024356937611, 10.83840613720542, 11.14232578857459, 11.70009518733754, 12.08862746568252, 12.58665309417458, 13.12948057558481, 13.38558974447904
