| L(s) = 1 | − 7-s + 2·11-s − 4·13-s + 7·17-s − 2·19-s + 8·23-s − 6·29-s + 4·31-s − 2·37-s − 41-s − 4·43-s − 7·47-s − 6·49-s − 6·53-s + 4·59-s + 4·61-s − 2·67-s − 8·71-s + 3·73-s − 2·77-s − 5·79-s + 6·83-s + 5·89-s + 4·91-s − 3·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s − 1.10·13-s + 1.69·17-s − 0.458·19-s + 1.66·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.156·41-s − 0.609·43-s − 1.02·47-s − 6/7·49-s − 0.824·53-s + 0.520·59-s + 0.512·61-s − 0.244·67-s − 0.949·71-s + 0.351·73-s − 0.227·77-s − 0.562·79-s + 0.658·83-s + 0.529·89-s + 0.419·91-s − 0.304·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−12.93024957602578, −12.71738933544998, −12.00298800526950, −11.82349500276427, −11.26088463014163, −10.73092230843509, −10.16748572573604, −9.782173881441006, −9.450412847373351, −8.945706352932181, −8.370253611307562, −7.852471961609558, −7.402642165202561, −6.905331487058079, −6.517660612475827, −5.917047203489062, −5.317621496620000, −4.955520321330430, −4.440200693655217, −3.674882467572029, −3.217703942905902, −2.878009606601774, −2.024831995522295, −1.460111608015848, −0.7909464142683819, 0,
0.7909464142683819, 1.460111608015848, 2.024831995522295, 2.878009606601774, 3.217703942905902, 3.674882467572029, 4.440200693655217, 4.955520321330430, 5.317621496620000, 5.917047203489062, 6.517660612475827, 6.905331487058079, 7.402642165202561, 7.852471961609558, 8.370253611307562, 8.945706352932181, 9.450412847373351, 9.782173881441006, 10.16748572573604, 10.73092230843509, 11.26088463014163, 11.82349500276427, 12.00298800526950, 12.71738933544998, 12.93024957602578
