| L(s) = 1 | − 7-s + 6·13-s + 4·17-s + 19-s − 2·23-s − 5·25-s − 6·29-s + 3·31-s − 7·37-s + 2·41-s − 4·43-s − 6·47-s − 6·49-s − 2·53-s − 10·59-s − 61-s − 3·67-s − 4·71-s + 3·73-s + 5·79-s + 14·83-s − 16·89-s − 6·91-s + 13·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.66·13-s + 0.970·17-s + 0.229·19-s − 0.417·23-s − 25-s − 1.11·29-s + 0.538·31-s − 1.15·37-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 0.274·53-s − 1.30·59-s − 0.128·61-s − 0.366·67-s − 0.474·71-s + 0.351·73-s + 0.562·79-s + 1.53·83-s − 1.69·89-s − 0.628·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.844218314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.844218314\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.89004471141708, −13.69235476655907, −13.29342352736920, −12.61358858195257, −12.14390838628369, −11.67483209644052, −11.07163013417879, −10.69450576220065, −10.07780774576248, −9.497979608071241, −9.227944078221969, −8.344926476956559, −8.086724873537360, −7.558433835668427, −6.724396177634521, −6.389446149916568, −5.671142144967163, −5.449604455735426, −4.509846022595826, −3.879616315850522, −3.384079181314210, −2.961128858523547, −1.777302401608994, −1.494499420885576, −0.4517557996375722,
0.4517557996375722, 1.494499420885576, 1.777302401608994, 2.961128858523547, 3.384079181314210, 3.879616315850522, 4.509846022595826, 5.449604455735426, 5.671142144967163, 6.389446149916568, 6.724396177634521, 7.558433835668427, 8.086724873537360, 8.344926476956559, 9.227944078221969, 9.497979608071241, 10.07780774576248, 10.69450576220065, 11.07163013417879, 11.67483209644052, 12.14390838628369, 12.61358858195257, 13.29342352736920, 13.69235476655907, 13.89004471141708
