| L(s) = 1 | − 2-s + 4-s + 3·5-s + 3·7-s − 8-s − 3·10-s + 6·13-s − 3·14-s + 16-s + 4·17-s + 4·19-s + 3·20-s − 7·23-s + 4·25-s − 6·26-s + 3·28-s − 10·31-s − 32-s − 4·34-s + 9·35-s − 6·37-s − 4·38-s − 3·40-s − 8·41-s + 6·43-s + 7·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.13·7-s − 0.353·8-s − 0.948·10-s + 1.66·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.670·20-s − 1.45·23-s + 4/5·25-s − 1.17·26-s + 0.566·28-s − 1.79·31-s − 0.176·32-s − 0.685·34-s + 1.52·35-s − 0.986·37-s − 0.648·38-s − 0.474·40-s − 1.24·41-s + 0.914·43-s + 1.03·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.70536833920090, −13.38000664901322, −12.67866830804164, −12.15885452209214, −11.63559203411364, −11.13224000758413, −10.79409441557599, −10.20406281596553, −9.840458919449736, −9.366200550574916, −8.736203203603529, −8.492059648170270, −7.703951228441474, −7.615807559988617, −6.711816007072742, −6.123097457400736, −5.859671222851044, −5.239776677608206, −4.904437770795475, −3.748631779573468, −3.551524935915903, −2.680953078421000, −1.782253733682473, −1.619521557208298, −1.185817645818367, 0,
1.185817645818367, 1.619521557208298, 1.782253733682473, 2.680953078421000, 3.551524935915903, 3.748631779573468, 4.904437770795475, 5.239776677608206, 5.859671222851044, 6.123097457400736, 6.711816007072742, 7.615807559988617, 7.703951228441474, 8.492059648170270, 8.736203203603529, 9.366200550574916, 9.840458919449736, 10.20406281596553, 10.79409441557599, 11.13224000758413, 11.63559203411364, 12.15885452209214, 12.67866830804164, 13.38000664901322, 13.70536833920090
