| L(s) = 1 | + 5·7-s + 2·11-s − 6·13-s + 8·17-s + 4·19-s − 3·23-s + 7·29-s + 4·31-s − 4·37-s − 5·41-s − 8·43-s + 47-s + 18·49-s − 2·53-s + 6·59-s + 61-s + 3·67-s − 6·71-s + 8·73-s + 10·77-s + 10·79-s + 83-s − 9·89-s − 30·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 0.603·11-s − 1.66·13-s + 1.94·17-s + 0.917·19-s − 0.625·23-s + 1.29·29-s + 0.718·31-s − 0.657·37-s − 0.780·41-s − 1.21·43-s + 0.145·47-s + 18/7·49-s − 0.274·53-s + 0.781·59-s + 0.128·61-s + 0.366·67-s − 0.712·71-s + 0.936·73-s + 1.13·77-s + 1.12·79-s + 0.109·83-s − 0.953·89-s − 3.14·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.260705725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.260705725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.87691461637533, −12.91324101645208, −12.23468886316829, −11.93185339303122, −11.78285621490710, −11.28999917507393, −10.34054049411163, −10.19165542125170, −9.773581977359396, −9.079123659171464, −8.432346684411510, −8.041748955814624, −7.678084948960850, −7.197123953929364, −6.648968897738321, −5.838452314868058, −5.273473532481474, −4.915068091501473, −4.607867761406085, −3.745287367728029, −3.211928028689018, −2.458650821721631, −1.845843630355566, −1.251898347600347, −0.6741148459160092,
0.6741148459160092, 1.251898347600347, 1.845843630355566, 2.458650821721631, 3.211928028689018, 3.745287367728029, 4.607867761406085, 4.915068091501473, 5.273473532481474, 5.838452314868058, 6.648968897738321, 7.197123953929364, 7.678084948960850, 8.041748955814624, 8.432346684411510, 9.079123659171464, 9.773581977359396, 10.19165542125170, 10.34054049411163, 11.28999917507393, 11.78285621490710, 11.93185339303122, 12.23468886316829, 12.91324101645208, 13.87691461637533
