| L(s) = 1 | + 7-s + 2·11-s + 2·13-s + 2·17-s + 19-s + 4·23-s + 2·29-s + 31-s − 3·37-s + 6·41-s − 43-s − 2·47-s − 6·49-s + 6·53-s − 8·59-s − 5·61-s + 4·67-s + 3·73-s + 2·77-s + 3·79-s + 14·83-s − 12·89-s + 2·91-s − 97-s + 6·101-s − 13·103-s + 12·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 0.371·29-s + 0.179·31-s − 0.493·37-s + 0.937·41-s − 0.152·43-s − 0.291·47-s − 6/7·49-s + 0.824·53-s − 1.04·59-s − 0.640·61-s + 0.488·67-s + 0.351·73-s + 0.227·77-s + 0.337·79-s + 1.53·83-s − 1.27·89-s + 0.209·91-s − 0.101·97-s + 0.597·101-s − 1.28·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.373080806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373080806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.174289771247430755437448694345, −7.49957345286182047133447815852, −6.72200660650854771203151507459, −6.07052435475033397076745602703, −5.24558835927647488468191831574, −4.53427211363349881726978703394, −3.66662024888744637241817137080, −2.92045371772590942200239372356, −1.75549660350439120776171169419, −0.880870392301658682999941370165,
0.880870392301658682999941370165, 1.75549660350439120776171169419, 2.92045371772590942200239372356, 3.66662024888744637241817137080, 4.53427211363349881726978703394, 5.24558835927647488468191831574, 6.07052435475033397076745602703, 6.72200660650854771203151507459, 7.49957345286182047133447815852, 8.174289771247430755437448694345
