| L(s) = 1 | − 2·5-s + 3·11-s + 13-s + 17-s − 3·19-s − 25-s − 3·29-s − 4·31-s − 2·37-s + 2·41-s − 6·43-s + 9·47-s − 53-s − 6·55-s − 11·59-s − 11·61-s − 2·65-s + 7·67-s + 15·71-s + 12·73-s − 2·79-s − 2·85-s + 10·89-s + 6·95-s + 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.904·11-s + 0.277·13-s + 0.242·17-s − 0.688·19-s − 1/5·25-s − 0.557·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s − 0.914·43-s + 1.31·47-s − 0.137·53-s − 0.809·55-s − 1.43·59-s − 1.40·61-s − 0.248·65-s + 0.855·67-s + 1.78·71-s + 1.40·73-s − 0.225·79-s − 0.216·85-s + 1.05·89-s + 0.615·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.09200377760903, −13.72782684296043, −12.97629307579325, −12.59401241852295, −12.02992763689183, −11.74976062232457, −11.08885101536511, −10.78929292141432, −10.23285191621118, −9.354399946808885, −9.239555794050744, −8.589043745245570, −7.871627017821642, −7.766259964004794, −6.927015087035190, −6.560379725786994, −5.949114882114035, −5.346119922436652, −4.666339220671715, −4.065919544989392, −3.678815882319450, −3.188352833883908, −2.227854036737112, −1.654541962859183, −0.8095137584680214, 0,
0.8095137584680214, 1.654541962859183, 2.227854036737112, 3.188352833883908, 3.678815882319450, 4.065919544989392, 4.666339220671715, 5.346119922436652, 5.949114882114035, 6.560379725786994, 6.927015087035190, 7.766259964004794, 7.871627017821642, 8.589043745245570, 9.239555794050744, 9.354399946808885, 10.23285191621118, 10.78929292141432, 11.08885101536511, 11.74976062232457, 12.02992763689183, 12.59401241852295, 12.97629307579325, 13.72782684296043, 14.09200377760903
