| L(s) = 1 | − 2·3-s − 7-s + 9-s − 6·13-s + 2·17-s + 8·19-s + 2·21-s − 2·23-s + 4·27-s − 4·31-s − 6·37-s + 12·39-s − 12·41-s + 4·43-s − 6·47-s + 49-s − 4·51-s − 14·53-s − 16·57-s − 8·61-s − 63-s + 6·67-s + 4·69-s − 8·71-s + 2·73-s + 12·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.417·23-s + 0.769·27-s − 0.718·31-s − 0.986·37-s + 1.92·39-s − 1.87·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.560·51-s − 1.92·53-s − 2.11·57-s − 1.02·61-s − 0.125·63-s + 0.733·67-s + 0.481·69-s − 0.949·71-s + 0.234·73-s + 1.35·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.46374568172881, −12.27953251493660, −11.96132815128410, −11.59345684783210, −10.95783139131228, −10.65790660756656, −9.960220819723958, −9.758607017994124, −9.404600573152038, −8.731493945136480, −8.109942654338218, −7.478659871575479, −7.373407909375796, −6.621798402999788, −6.376065512856133, −5.642396772538724, −5.235602488658775, −5.019802404195019, −4.515143911083056, −3.606002151804172, −3.226276114012088, −2.733857250185117, −1.889947720458155, −1.368770831447063, −0.5142744417660891, 0,
0.5142744417660891, 1.368770831447063, 1.889947720458155, 2.733857250185117, 3.226276114012088, 3.606002151804172, 4.515143911083056, 5.019802404195019, 5.235602488658775, 5.642396772538724, 6.376065512856133, 6.621798402999788, 7.373407909375796, 7.478659871575479, 8.109942654338218, 8.731493945136480, 9.404600573152038, 9.758607017994124, 9.960220819723958, 10.65790660756656, 10.95783139131228, 11.59345684783210, 11.96132815128410, 12.27953251493660, 12.46374568172881
