| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s + 11-s + 4·13-s + 2·15-s + 4·17-s − 2·19-s + 4·21-s − 4·23-s − 25-s + 27-s + 4·29-s + 33-s + 8·35-s + 37-s + 4·39-s − 2·41-s − 6·43-s + 2·45-s + 8·47-s + 9·49-s + 4·51-s + 2·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.970·17-s − 0.458·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.174·33-s + 1.35·35-s + 0.164·37-s + 0.640·39-s − 0.312·41-s − 0.914·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.274·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.176192045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.176192045\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.53370248346666, −15.05241965196753, −14.46128212132581, −14.07670722446387, −13.64691490880862, −13.23429857677970, −12.26579078579839, −11.95420902771785, −11.18343647994095, −10.69212452593158, −10.07230489587931, −9.603388602418153, −8.773163661096347, −8.392987198593270, −7.948693842465693, −7.242947713499192, −6.433658623007606, −5.816249960581098, −5.314288761369826, −4.470264418295071, −3.937942302646532, −3.107855682041938, −2.158063030544045, −1.662561348544483, −1.001675148588481,
1.001675148588481, 1.662561348544483, 2.158063030544045, 3.107855682041938, 3.937942302646532, 4.470264418295071, 5.314288761369826, 5.816249960581098, 6.433658623007606, 7.242947713499192, 7.948693842465693, 8.392987198593270, 8.773163661096347, 9.603388602418153, 10.07230489587931, 10.69212452593158, 11.18343647994095, 11.95420902771785, 12.26579078579839, 13.23429857677970, 13.64691490880862, 14.07670722446387, 14.46128212132581, 15.05241965196753, 15.53370248346666
