| L(s) = 1 | + 2·5-s − 2·7-s + 3·13-s + 17-s − 5·19-s − 25-s − 6·29-s − 4·35-s − 6·37-s − 4·41-s − 11·43-s − 9·47-s − 3·49-s + 6·53-s + 4·59-s − 4·61-s + 6·65-s − 9·67-s − 8·71-s + 16·73-s − 10·79-s + 9·83-s + 2·85-s − 3·89-s − 6·91-s − 10·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.832·13-s + 0.242·17-s − 1.14·19-s − 1/5·25-s − 1.11·29-s − 0.676·35-s − 0.986·37-s − 0.624·41-s − 1.67·43-s − 1.31·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s − 0.512·61-s + 0.744·65-s − 1.09·67-s − 0.949·71-s + 1.87·73-s − 1.12·79-s + 0.987·83-s + 0.216·85-s − 0.317·89-s − 0.628·91-s − 1.02·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5468298247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5468298247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.98373814355827, −12.20896629182363, −11.87148156799211, −11.25990806253597, −10.82187091671664, −10.30811993458566, −9.960811651682462, −9.564991175110986, −9.030090262956035, −8.587974125558494, −8.190694644216024, −7.578814675409656, −6.867267720568328, −6.564682050884322, −6.164249740839873, −5.659418053616421, −5.198459034498576, −4.647664001902727, −3.809062929102190, −3.601873215547870, −2.964450252298464, −2.292399388590819, −1.692906928400175, −1.361975782007840, −0.1838894372862091,
0.1838894372862091, 1.361975782007840, 1.692906928400175, 2.292399388590819, 2.964450252298464, 3.601873215547870, 3.809062929102190, 4.647664001902727, 5.198459034498576, 5.659418053616421, 6.164249740839873, 6.564682050884322, 6.867267720568328, 7.578814675409656, 8.190694644216024, 8.587974125558494, 9.030090262956035, 9.564991175110986, 9.960811651682462, 10.30811993458566, 10.82187091671664, 11.25990806253597, 11.87148156799211, 12.20896629182363, 12.98373814355827
