| L(s) = 1 | − 2·4-s − 7-s + 4·13-s + 4·16-s − 17-s + 2·19-s + 9·23-s + 2·28-s + 9·29-s + 5·31-s − 2·37-s + 43-s − 9·47-s + 49-s − 8·52-s − 9·59-s − 61-s − 8·64-s − 5·67-s + 2·68-s + 9·71-s + 7·73-s − 4·76-s − 10·79-s + 9·89-s − 4·91-s − 18·92-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 1.10·13-s + 16-s − 0.242·17-s + 0.458·19-s + 1.87·23-s + 0.377·28-s + 1.67·29-s + 0.898·31-s − 0.328·37-s + 0.152·43-s − 1.31·47-s + 1/7·49-s − 1.10·52-s − 1.17·59-s − 0.128·61-s − 64-s − 0.610·67-s + 0.242·68-s + 1.06·71-s + 0.819·73-s − 0.458·76-s − 1.12·79-s + 0.953·89-s − 0.419·91-s − 1.87·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.09268639880606, −13.68647498220987, −13.29864677816504, −12.88063814251066, −12.36400259563770, −11.79703843872922, −11.25612824929337, −10.54838033889979, −10.37794988932413, −9.524740132718236, −9.266740935202754, −8.699537012952535, −8.257797177670553, −7.812923493483532, −6.940304826852722, −6.541559377129955, −6.000195452925522, −5.276433991711893, −4.804524596156897, −4.365073699924622, −3.529585451180649, −3.188545302128726, −2.538754303024195, −1.263704225060293, −1.030725376342124, 0,
1.030725376342124, 1.263704225060293, 2.538754303024195, 3.188545302128726, 3.529585451180649, 4.365073699924622, 4.804524596156897, 5.276433991711893, 6.000195452925522, 6.541559377129955, 6.940304826852722, 7.812923493483532, 8.257797177670553, 8.699537012952535, 9.266740935202754, 9.524740132718236, 10.37794988932413, 10.54838033889979, 11.25612824929337, 11.79703843872922, 12.36400259563770, 12.88063814251066, 13.29864677816504, 13.68647498220987, 14.09268639880606
