| L(s) = 1 | − 2·7-s + 11-s + 2·13-s − 2·17-s − 6·19-s + 4·23-s − 5·25-s − 10·29-s + 8·31-s + 6·37-s − 6·41-s + 6·43-s − 3·49-s + 12·53-s + 12·59-s + 6·61-s + 8·67-s + 8·71-s − 10·73-s − 2·77-s − 10·79-s + 4·83-s + 16·89-s − 4·91-s − 14·97-s − 2·101-s − 4·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 1.37·19-s + 0.834·23-s − 25-s − 1.85·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.914·43-s − 3/7·49-s + 1.64·53-s + 1.56·59-s + 0.768·61-s + 0.977·67-s + 0.949·71-s − 1.17·73-s − 0.227·77-s − 1.12·79-s + 0.439·83-s + 1.69·89-s − 0.419·91-s − 1.42·97-s − 0.199·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505220082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505220082\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.137877385419272587734483224647, −7.20300503359676768568111986003, −6.58278382469561048632165262703, −6.03678412878931232508658806687, −5.27000959062313134340258385299, −4.16510007625915039939928222095, −3.79830622205186791501117577874, −2.72434347711442386925162874595, −1.91957125953450629528217986656, −0.62339616520112493051272374163,
0.62339616520112493051272374163, 1.91957125953450629528217986656, 2.72434347711442386925162874595, 3.79830622205186791501117577874, 4.16510007625915039939928222095, 5.27000959062313134340258385299, 6.03678412878931232508658806687, 6.58278382469561048632165262703, 7.20300503359676768568111986003, 8.137877385419272587734483224647
