| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 13-s − 4·14-s + 16-s + 6·19-s + 20-s + 4·23-s + 25-s − 26-s − 4·28-s + 10·29-s − 4·31-s + 32-s − 4·35-s − 4·37-s + 6·38-s + 40-s + 6·41-s − 4·43-s + 4·46-s + 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.85·29-s − 0.718·31-s + 0.176·32-s − 0.676·35-s − 0.657·37-s + 0.973·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.589·46-s + 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.228718664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.228718664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.38687148161168, −13.00490227763787, −12.42842950556342, −12.19246717515803, −11.63578899734157, −11.03410894936174, −10.43125947436417, −10.04384679659930, −9.698972154681593, −8.953026678064456, −8.840467391054776, −7.847222708655032, −7.285656802079353, −6.950576108823446, −6.428533436998815, −5.890525942004218, −5.524034504552873, −4.840318749118323, −4.420522573725937, −3.471127424115515, −3.272734362109082, −2.718402366262236, −2.127709971551402, −1.172653436640661, −0.5795365291957699,
0.5795365291957699, 1.172653436640661, 2.127709971551402, 2.718402366262236, 3.272734362109082, 3.471127424115515, 4.420522573725937, 4.840318749118323, 5.524034504552873, 5.890525942004218, 6.428533436998815, 6.950576108823446, 7.285656802079353, 7.847222708655032, 8.840467391054776, 8.953026678064456, 9.698972154681593, 10.04384679659930, 10.43125947436417, 11.03410894936174, 11.63578899734157, 12.19246717515803, 12.42842950556342, 13.00490227763787, 13.38687148161168
