| L(s) = 1 | + 3-s − 5-s + 9-s − 5·11-s − 13-s − 15-s + 5·17-s − 4·19-s + 3·23-s + 25-s + 27-s + 8·29-s − 9·31-s − 5·33-s − 8·37-s − 39-s − 5·41-s + 43-s − 45-s + 8·47-s − 7·49-s + 5·51-s + 13·53-s + 5·55-s − 4·57-s − 8·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 1.21·17-s − 0.917·19-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.61·31-s − 0.870·33-s − 1.31·37-s − 0.160·39-s − 0.780·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.700·51-s + 1.78·53-s + 0.674·55-s − 0.529·57-s − 1.04·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−14.99294745752644, −14.57731190455502, −13.98216573690953, −13.46377185810703, −12.89935330039109, −12.47379304046930, −12.06074115930840, −11.33509840845140, −10.59392824467277, −10.34475602445738, −9.923800034154028, −8.891546910541076, −8.745109093010952, −8.051576957677062, −7.488504651966982, −7.229830254674500, −6.410849337609021, −5.645232791750276, −5.045053331931768, −4.652179274655550, −3.650445456290571, −3.316595933280144, −2.541076993952438, −1.989469634629209, −0.9408085097086323, 0,
0.9408085097086323, 1.989469634629209, 2.541076993952438, 3.316595933280144, 3.650445456290571, 4.652179274655550, 5.045053331931768, 5.645232791750276, 6.410849337609021, 7.229830254674500, 7.488504651966982, 8.051576957677062, 8.745109093010952, 8.891546910541076, 9.923800034154028, 10.34475602445738, 10.59392824467277, 11.33509840845140, 12.06074115930840, 12.47379304046930, 12.89935330039109, 13.46377185810703, 13.98216573690953, 14.57731190455502, 14.99294745752644
