| L(s) = 1 | + 5-s + 4·7-s + 2·11-s − 8·23-s + 25-s + 2·29-s + 2·31-s + 4·35-s + 8·37-s + 2·41-s + 4·43-s + 9·49-s − 6·53-s + 2·55-s + 14·59-s + 14·61-s − 4·67-s + 8·71-s − 10·73-s + 8·77-s − 6·79-s + 12·83-s − 14·89-s + 18·97-s − 10·101-s + 16·103-s − 14·109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.603·11-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.676·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s + 0.269·55-s + 1.82·59-s + 1.79·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.911·77-s − 0.675·79-s + 1.31·83-s − 1.48·89-s + 1.82·97-s − 0.995·101-s + 1.57·103-s − 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.872913826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872913826\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.131594756834447502080348823939, −7.56151588258959383908652429756, −6.63664175394745146065158701964, −5.93399176588351621916952557993, −5.25298500002688243277781975342, −4.42832903326511711184997057727, −3.89664727381198336373592494897, −2.56561855381752511488181703499, −1.85829193607977348011558635044, −0.956960668400174358503119313971,
0.956960668400174358503119313971, 1.85829193607977348011558635044, 2.56561855381752511488181703499, 3.89664727381198336373592494897, 4.42832903326511711184997057727, 5.25298500002688243277781975342, 5.93399176588351621916952557993, 6.63664175394745146065158701964, 7.56151588258959383908652429756, 8.131594756834447502080348823939
