| L(s) = 1 | − 3-s + 3·5-s + 2·7-s − 2·9-s + 11-s − 13-s − 3·15-s − 2·21-s − 3·23-s + 4·25-s + 5·27-s + 6·29-s + 31-s − 33-s + 6·35-s + 7·37-s + 39-s − 6·41-s + 8·43-s − 6·45-s + 12·47-s − 3·49-s + 6·53-s + 3·55-s − 9·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 0.774·15-s − 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s + 0.179·31-s − 0.174·33-s + 1.01·35-s + 1.15·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.404·55-s − 1.17·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206492 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206492 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.450675731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.450675731\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−13.04143410679471, −12.39815621608600, −12.16693324480664, −11.59284926252028, −11.15835918243218, −10.71243309083865, −10.21725194270925, −9.849612517412116, −9.268208446486938, −8.841102180612647, −8.348999666111287, −7.824315382546568, −7.261388412032722, −6.574740827316901, −6.114571268757061, −5.913894598311964, −5.209469399923577, −4.944290187636797, −4.317742043186739, −3.674336767728941, −2.774409856596078, −2.417287228221147, −1.865144259728538, −1.110680225735760, −0.5896519763778879,
0.5896519763778879, 1.110680225735760, 1.865144259728538, 2.417287228221147, 2.774409856596078, 3.674336767728941, 4.317742043186739, 4.944290187636797, 5.209469399923577, 5.913894598311964, 6.114571268757061, 6.574740827316901, 7.261388412032722, 7.824315382546568, 8.348999666111287, 8.841102180612647, 9.268208446486938, 9.849612517412116, 10.21725194270925, 10.71243309083865, 11.15835918243218, 11.59284926252028, 12.16693324480664, 12.39815621608600, 13.04143410679471
