| L(s) = 1 | − 5-s + 7-s + 3·11-s − 2·13-s − 17-s − 5·19-s − 8·23-s − 4·25-s + 3·29-s − 4·31-s − 35-s + 8·37-s + 2·41-s + 12·43-s − 4·47-s + 49-s + 6·53-s − 3·55-s + 14·59-s − 10·61-s + 2·65-s − 15·67-s − 71-s + 14·73-s + 3·77-s + 10·79-s + 14·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s − 0.554·13-s − 0.242·17-s − 1.14·19-s − 1.66·23-s − 4/5·25-s + 0.557·29-s − 0.718·31-s − 0.169·35-s + 1.31·37-s + 0.312·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.404·55-s + 1.82·59-s − 1.28·61-s + 0.248·65-s − 1.83·67-s − 0.118·71-s + 1.63·73-s + 0.341·77-s + 1.12·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441708109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441708109\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.54833020659755, −14.04541264972997, −13.55751873919542, −12.89028949696415, −12.29289217908313, −11.98571828923930, −11.50279090423334, −10.86175078164328, −10.49981450753775, −9.696344839932115, −9.368634016847940, −8.721974833044816, −8.087261280133763, −7.764408679142260, −7.143457312040837, −6.406034763003355, −6.062071711283339, −5.367520268370120, −4.546204422922526, −3.998061012617921, −3.865694156734276, −2.627331709293041, −2.194385413528403, −1.398750260926403, −0.4221072572761915,
0.4221072572761915, 1.398750260926403, 2.194385413528403, 2.627331709293041, 3.865694156734276, 3.998061012617921, 4.546204422922526, 5.367520268370120, 6.062071711283339, 6.406034763003355, 7.143457312040837, 7.764408679142260, 8.087261280133763, 8.721974833044816, 9.368634016847940, 9.696344839932115, 10.49981450753775, 10.86175078164328, 11.50279090423334, 11.98571828923930, 12.29289217908313, 12.89028949696415, 13.55751873919542, 14.04541264972997, 14.54833020659755
