| L(s) = 1 | + 4·5-s + 3·7-s + 11-s − 13-s + 17-s + 4·19-s + 2·23-s + 11·25-s − 29-s − 10·31-s + 12·35-s − 6·37-s − 6·41-s − 4·43-s + 3·47-s + 2·49-s + 4·53-s + 4·55-s + 10·59-s + 14·61-s − 4·65-s + 7·67-s − 2·71-s − 2·73-s + 3·77-s + 16·79-s + 2·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.13·7-s + 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.417·23-s + 11/5·25-s − 0.185·29-s − 1.79·31-s + 2.02·35-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.437·47-s + 2/7·49-s + 0.549·53-s + 0.539·55-s + 1.30·59-s + 1.79·61-s − 0.496·65-s + 0.855·67-s − 0.237·71-s − 0.234·73-s + 0.341·77-s + 1.80·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.277684534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.277684534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−16.06551801035070, −14.99477984300909, −14.82188286231801, −14.07713831562008, −13.90968004328171, −13.21918333567437, −12.69179877491698, −12.01873145688475, −11.30963487853634, −10.91003516026979, −10.04710507769737, −9.864574815617214, −9.049789141754306, −8.687522212964355, −7.901922940772204, −7.052406993714588, −6.746448268677296, −5.711856794155373, −5.291944547765907, −5.041190249063762, −3.921868026202518, −3.111785572687455, −2.103585905507431, −1.774404277253295, −0.9303146426709842,
0.9303146426709842, 1.774404277253295, 2.103585905507431, 3.111785572687455, 3.921868026202518, 5.041190249063762, 5.291944547765907, 5.711856794155373, 6.746448268677296, 7.052406993714588, 7.901922940772204, 8.687522212964355, 9.049789141754306, 9.864574815617214, 10.04710507769737, 10.91003516026979, 11.30963487853634, 12.01873145688475, 12.69179877491698, 13.21918333567437, 13.90968004328171, 14.07713831562008, 14.82188286231801, 14.99477984300909, 16.06551801035070
